OP I'm somewhat in your situation* and will echo what others are saying: don't buy together yet. The relationship is too new and there's a risk that, if things go badly, you'll be financially, legally and physically attached to your SO while dealing with separating your joint property and assets.
In your case, since it seems you want to keep your current house, having your SO move in with you and pay rent and split utilities would be the best bet. You save, your SO saves, you get to live together without and legal or financial obligations. In time, if all goes well, you can then decide if and how you want to share housing.
*SO and I are not married but we've been together a fairly long time. We have no kids from past relationships or prior financial obligations to our ex-spouse's. We both owned our seperate homes. Our choices when we decided to live together were:
1). one sells and moves in with the other as a "renter" with no legal claim to the house
2). One sells and moves in with the other and pays them 1/2 of the current house value (no mortgage) and gets added to the deed.
3). We both sell and rent together and invest our sale proceeds separately and keep finances separate.
4).We both sell and buy a place together with 50/50 ownership.
5). We rent out one or both places and live together (buy or rent) in another place we get together.
We are currently doing #3 but plan, eventually, plan to do #4. There are some issues to #4 if you are not married such as how to hold title, who gets the property if one dies and want to leave their share to someone else, or if one wants to sell and the other doesn't, if one amassed huge bills from a law suit or medical problem and they consider your property joint property and come after your share of the equity, etc. So owning property together after such a short term relationship may have longer term financial issues you'd need to discuss and work out.