Hello OP
I'm very much a cat person, my DP was not, having grown up with pet dogs.
When I moved into my first home I was asked to foster a litter of 5, who never left. They all lived to a very good age (first went at 16, last at 21). I was devastated. The cats were 8 when my DS was born, he'd grown up alongside them.
I got together with my now DP around the time I was cat-free. It was around 6 months later I felt the need to adopt again. I made sure my DS was on board; cats of any age can be very destructive until they settle into their new home. Apologies if I missed it, I can't see that you've said how old your cats are? Advice will differ according to age.
I got 2 sisters from the local rescue centre. The cattery part is at the back of the kennels, poor little things were terrified of the barking. In fact they were terrified full stop. I'd never known such frightened cats (they were 12 weeks old).
Got them home, barely saw them for a month as they were always hiding. Cats are crepuscular (might not have spelled that right) which means they are most active at dawn and dusk. This is the best time to play with them. Provide safe hiding spaces at different heights if you can, so your cats have somewhere to bolt to if they are feeling overwhelmed. You can get cheap cardboard cat scratchers impregnated with catnip; I've put one in most rooms and on the landing. I can second clear adhesive tape on areas you don't want them to scratch because they hate the sticky sensation on their paws. Once the habit of using that spot is broken, they won't go back. You can also try sprinkling out-of-bounds areas (such as stair carpets or the arms of sofas) with citrus oils because the majority of cats detest the taste.
My girls don't go outside at all and never have. They don't even try when the bi-folds are wide open in the summer. You can really, really help with the litter tray smell by feeding a high quality diet. Cats don't need cereals or vegetables, which are often used to bulk out a lot of the most popular brands of cat food. Their digestive systems are not designed to process cereals or vegetables, hence the awful smell at the other end! Switch to a brand that contains the highest quantity of meat/poultry/fish you can.
Your outdoor cat will eat grass to make themselves sick to bring up furballs. If you want to avoid your mostly indoor cat vomiting indoors, brush her regularly (mine love the mini Furminator brush) and provide an anti furball dry/biscuit food. It contains enzymes which break down the fur in their stomach, meaning it passes through rather than back up. Always make sure they have access to fresh drinking water.
When my DP moved in, he knew my cats were absolutely non-negotiable. At first he was like your DH, i.e. ignored them. 2 years later he adores them, and they adore him back. As he started spending more time around them/giving them attention, he noticed mild allergy symptoms. One antihistamine a day sorts that out completely! DP even talks to the cats in a silly voice when he thinks no-one can hear
I would never countenance getting rid of my cats, but I would do everything I could to ensure they were as happy as possible. By doing that, I'm sure the issues your DH is having will alleviate quite quickly.
Good luck