After your last post, I remembered that DH bought the house we live in before he and BM were divorced, and it never came up, at all.  I don't think it's an issue at all - if BM had a claim in it, she'd have gone for it, you'd better believe. We are in NY.

First get pre-qualified at your bank or a mortgage broker - that means they will tell you how much you can afford based on your income. Look for houses that are less than what they pre-qualify you for, it's crazy how much they think is OK to pay for a mortgage.  Your realtor will send you listings and arrange viewings for you with the realtor who is selling the house.  It's not complicated, they will take care of everything for you - setting up inspections, attorney for closing, etc. If you don't think the one you pick is doing what they need to do, fire them and find another.

Good luck. My DH moved his stuff slowly to an apartment and then left entirely when BM was out of town. He too feared she would be vindictive or vengeful.

This way you don't waste your time or the realtor's looking at houses you clearly KNOW you are not interested in.

I would shop around for mortgage interest rates. There are lots of great incentives for first time buyers.

Also, I would explore looking for sellers who are selling ourtright,(for sale by owner) and not going through a realtor. So much of the money is spent on realtor fees. Often you can find highly motivated sellers who may even reduce the price significantly if they want to get out from under the mortgage payment.

Also, consider city taxes, opposed to living outside the city limits. I know here the taxes go up exponentially higher each and every year.

Since your marriage was so brief (I think 2 years if memory serves correctly) you will not have a problem buying a home solely in your name. Do not list HIM ANYWHERE on the loan application.

As mentioned previously, look into pet daycare options, or classes while you're away. I know you know better than to leave them in that environment.

When you start to sell things, he will become suspicious, so it's very important you get all your valuables and sentimental items out FIRST. Put photos, important papers and jewelry, (for example) in a safety deposit box. 

Have you spoken to a divorce atty?  Each state is different but the vast majority will consider a real estate purchase made prior to separation to have been made with marital funds.  He very well may have a claim on anything you purchase.  The suggestion above of renting a house, which are more likely to allow pets, is a good one.  I would focus my efforts on that rather than making a purchase with all of the attendant risks.

Do not even consider buying property without talking to a lawyer to find out how that property will be affected in a divorce.  One thing to be aware of is that if you are married and you buy a home solely in your name, the mortgage company may as part of the closing ask your husband to quitclaim his interest in your house or otherwise sign something acknowledging he has no interest.  If you are trying to keep the fact that you're buying your own house and planning to file for divorce from him, that would spill the beans.  Your husband could also refuse to sign.  So ask an attorney if that would be necessary, and ask your realtor and any potential lender about it ASAP.  

You might do better to look for an apartment or a rental house that allows pets for now, and buy the house after the divorce is final.  Before I owned my house, I rented with 2 dogs (45 pounds and 70 pounds), and while I was very limited in where I could rent, I did, after a lot of looking and some negotiation, find a place that would allow my pets.  

Store everything else in storage unit.  Bimmy bammy boom...you can be out in one day while he is at work.  
Or look into apartment rental, air bnb, house sitting gigs. Give a damage deposit to assure the owners you will be responsible for the pets.  
 

also look at Facebook marketplace for selling items, and the local online Facebook yard sale sites.  Donate the rest to thrift shops. It is just stuff...keep only that which give you joy and you can move without too much hassle.  Adopt a minimalist zen attitude to go with your new free life.  Less stuff equals more cuddle time with the dogs.  Take up dog sports, training, hiking clubs. 
 

start your new life sooner than later.  No one ever says they wished they stayed in a bad situation longer.  Everyone always says they wished they had not wasted their life in a bad place.  
Go. Just go.  

I agree first step consult a divorce lawyer... know those issues first.

When you are ready to buy...a few things to think about.

1. Ask friends and coworkers for realtor recommendations...or at least online reviews. You do NOT have to sign a buyer's agent agreement..if you dont care for your realtor... find a different one.

2. For sale by owners are sometimes a decent deal...but I find they often are unreasonably priced..owners attach emotion in their pricing sometimes..sure some are trying to save the listing fees.. but depending upon where you live... paperwork can be complicated. 

A realtor can often direct you to a mortgage broker...who can shop rates... cap center and quicken also have good products.  I've had worst luck going with my banks for some reason.

If you want to throw your H off the scent, pretend that you are jumping on the minimalist bandwagon. Buy Marie Kondo's book, make sure he sees it, and talk up decluttering. Get a mail drop and a storage unit. Start going through closets and offer to help him sort through his wardrobe so you can donate unwanted goods to a charity shop, all as a cover for removing valuables. Let him think you're fired up with home improvement zeal. It's amazing how much you can sneak out in a trunk.

As for home buying, google Realtors in _______, NY and find one who is well seasoned, preferably female. Interview several before deciding which you feel comfortable with, and find one that has a track record of actually hustling for their clients.

Buying a house isn't rocket science.  You can check ratings on agents.

A buyers agent is paid by the seller based on the sales price of the home.  Usually 6% of the sales price is split between the sellers agent and the buyer's agent.  Your agent can recommend mortgage brokers.  The broker does not ultimately matter all that much since many mortgages are sold several times during the course of the mortgage term.  Interest rates are a factor of the market and your credit rating.  You will have to come up with your down payment and closing costs.

The title company will research the title to ensure that it is clear and unencumbered by liens.  The title policy protects you.

Don't be intimidated by the process.  It is well defined, you are protected by applicable law and the Realtors, title company and mortgage brokers all want it all to go smoothly.

Don't over complicate it.  Research who you want to sign a buyer's agent agreement with and they will step you through the purchase process.  That is what they do.

Someone recommended that you look into an RV to live in with your pets.  You can rent them and place them at a pet friendly RV park.  I think that recommendation is a good idea so you can get you and your pests out quickly. Rent a secure monitored storage facility for your stuff and have it moved.

There is no need to overcomplicate any of this.

Take care of you.

I am not a lawyer, but acquiring assets while still married, esp a house, is not in your best interest. I understand that its tough finding  a place that takes big dogs (and yet is required to take children, who can be just or more destructive...) but get that storage space, and gradually move things there. Im sending you digital hugs and prayers that you can get out very soon.

This way you can take your time. Buying a home is hardwork and a process - it takes time, and you dont want to rush it.