You will have to decide if this is something you can live with, or not. 

My formerSO's kid was nasty like this. He was a TEEN and ate with his hands... everything. Spaghetti, salad (which was DRENCHED in creamy dressing), vegetables (peas, beans, corn off the cob), meat (that his daddeeee had cut for him). Would scoop peanut butter out of the jar with his fingers and put it in his mouth...and then put his sucked off fingers back in the jar!!! 

Seriously, this would make me literally freaking gag. Also, would not wash his hands after using the restroom...so added grossness. Of course "I was over reacting".

His dad and the rest of the family thought it was "cute"..."he's just a kid"..."his mom didn't teach him any better" (of course this was after his dad had custody for YEARS, but still bm fault). Also apparently it became *my* job to teach him somehow - erm, hard pass, especially when family wouldn't enforce? The rest of the family did not behave this way....not sure why they allowed him to be feral...

This teen was NOT "overall a good kid". I left the relationship for many reasons, this feral, mean, thief of a teen was one of them.

Can you learn to live with it? That is literally the only thing you will be able to change. If not, start planning your exit. 

I was raised similar. I literally did not know how to use utensils correctly. My ex DH taught me on our first date. It's sad really, I was so incredibly self conscious and knew I was lacking in social skills. I think this is additionally what set me up for a dysfunctional toxic marriage to the same man. That said, table manners as well as other common courtesies became very important to me when raising my own children. Kids won't just wake up one day and have them. They must be taught over time. If you have good communication with your partner, I would discuss the disadvantages of not having these manners. Trust me I have a 38 year old step son that refuses to use a fork the majority of time. He claims a multitude of "reasons" I think it's just laziness. 

Yeah my SDs used their hands a lot, alternating with a spoon. It was a revelation when, at 8 & 10 I started insisting on proper use of utensils. And yes, hand washing! Key: DH supported this change. If your SO doesn't agree you will get undermined and will be the ogre. Talk with your SO first and if you get resistance there stop.

If you cannot solve this then you can insist on your personal space....no dirty kids in bed (they shouldn't be there anyway!!!!!.....or on the couch. No kids helping with meals unless you see them wash their hands. No going out to eat with them for same reasons. 

Particularly my niece.

My brother and I were raised with very strictly enforced table and interpersonal manners.

Our parents worked for the State Department and we often were at Embassy functions growing up.

So, my brother's kids horrendous table manners has always been mind boggling to me.

My parents were shocked. To the point that my dad would not tolerate it at his table.  The memorable incident was when my niece was about 7-9yo and everyone was at mom and dads for a holiday.  My niece sat at the table with her feet crossed in the chair eating with her fingers.  He told her to sit up straight, put her napkin in her lap, use the eating utensils.  My brother was silent, my SIL was trying to not have my niece have hurt fee fees. SIL was all 'Oh, its okay. Grampa is not really mad at you.'  My dad corrected SIL in front of eveyrone ans said that it is not okay and if my niece continued he would move her plate next to the dog bowl and she could eat there since that is how she was behaving.

My brother had to give his wife the look to get her to not escalate as my dad stipulated the basic manners requried.

As the dinner progressed my niece either lost her mind or was testing boundaries thinking that her mommy would coddle her.  She slouched back in her chair and put her feet up on the table on either side of her plate.  My did, not missing a beat of a bite of his diner, picked up his table knife by the blade and whacked her on the ankle with the handle of the knife.

Never again did not brother's kids use anything but acceptable table manners at a family event. Nor anywhere else I have ever had a meal with them.  Their dad is a Sr. corporate executive, though he was not at the time of the ankle whack,  and over the years they were involved in a large number of formal events.

If my brother had maintained the standards he was raised with, this event and other table manner events would not have happened.

If kids are table animals, tell them to leave the table.

BOTH my skids had bad manners, in general. SD23 Feral Forger used to burp and fart openly and laugh. She would leave her period pads all open in the garbage can.

Sd16 power sulk - shes generally a lot better. My only complaint is that she slurps and sucks her finger and eats with her hands and doesnt wash.

Its not going to get better unless they are taught better.

My SD19 went through a phase where she thought she was a nature girl and never washed her hands or wanted to bathe. She would show us Youtube articles by random hippie wierdos that said bathing is BAD for you, blah blah blah. We just laughed and told her she still needs to brush her teeth, wash her hands, etc. 

Then she found wriggling worms in her poo in the toilet one day, and FREAKED the f*** out. Thankfully it was at her BM's. BM took her to the doctor and I think she had hookworms from playing in the dirt outside, walking barefoot, and not washing up her hands afterwards. She had to take a dewormer.  I thought it was disgusting but kind of funny, since she was such a NATURE GIRL. She is fine now, and I notice she is much better about washing her grubby hands.