I have been exactly where you are.
I would not recommend selling your house to move in with SO at all.
Why would you want to give up your ability to be independent in order to put your security at the mercy of someone who has no legal responsibility to you? If your SO owns the house you would be moving into, then he has the right to kick you out on the street at any time. Now of course right now you think that would never happen- but it could. Then you would have no place to go. Or what if you decided after a year or two that it wasn't working out with your SO? You could potentially be stuck there because the costs of moving out might be more than you can afford.
DO NOT put yourself in the position of being dependent on anyone else for something as basic as housing. Ever.
You cannot- as far as I know- just get put onto someone else's mortgage. SO would have to refi and they would have to write the mortgage in both names. But you don't want to ever own a house with someone you're not married to. Way too much risk there, should the relationship go sour.
Renting your house out? No matter how "nice" the renters are, they will not treat your home as anything other than a rental. I could tell you horror stories about what I found at my house after the renters left, and what it cost me to clean it up and attempt to repair it all (it was thousands of $$). One of the bedrooms STILL stinks like their gross smelly teenage son, 3 1/2 years later. I have tried everything to get the stink out of that room. They had a pitbull in my house, on a lease that prohibited dogs. I ended up having to sue her for costs over and above the security deposit, but I'll never see a dime.
But in the end, I am so glad I did not sell my house when I moved in with then-SO. When the whole thing went south, I was able to move back into MY home relatively easily. I had options, and a place to go that was MINE and no one could take away from me. Did I mind having to kick the tenants out? No way- that's what renting is- temporary.
Your SO may love you, but when it comes to his financial investment in his home and his financial responsibilities- he would be an idiot to put that in jeopardy. He will make decisions based on what's best for HIM, and he's right to do that. But it won't be what's best for you.
Keep your house, make the fixes as you can, and then sell- but only if it's what's best for YOU. If you can't sell without sinking a shit ton of money that you don't have into the place- then you're not in a position to sell.
You will never regret protecting yourself financially and ensuring that you can put a roof over your head without anyone else's help.
I regret moving in with that SO and renting out my house every day. Don't be me.