The problem isnt if Section 8 allows pets. the problem is if a landlord will allow pets. Most apartments only allow two of a certain weight and restrict some breeds. Also, I noted you said that you had a lot of money saved up. Money dwindles pretty quickly and if you are applying for Section 8, it means that you are not in a position where money is going to replenish itself as quickly as it did for you in the past. Pets are pretty expensive and accommodating them is going to be your biggest uphill battle. Please, if you can, once you apply for section 8, make sure that you have a plan in place to rehome any pets that don’t meet landlord standards. Also, please do not take on any more pets. I know people do that when they are in stressful situations. Also, do not consider trying to sneak a pet in, especially a large dog. If your apartment even allows large breed dogs, you are going to be pushing your luck with your neighbors with just two. None of them would put up with three.
My only real suggestion would be to find a house to rent. But Section 8 isn’t going to give you a decent voucher for a house if it’s just one person. You’d need a family of people (not pets) to get that kind of voucher. If you want to keep your big dogs, you are probably going to be paying a private homeowner to rent you their property at market rate. You may be able to find one if you look far enough out into the country.