Not sure if I’m in a position to buy a house, need advice.

My girlfriend and I are trying to figure out whether we’re in a good position to buy our first house or if we should continue renting and saving for another year or two.

1. I’m 25 and make $97k as a first-year engineer.
2. My girlfriend is 23 and makes about $38k working at the largest hospital in our county.
3. Combined gross income is about $135k/year.
4. I have about $15k in student loans. She has about $10k in student loans.
5. Both of our cars are completely paid off. No credit card debt.
6. We currently have about $44k combined in savings.
7. Our rent + utilities is around $2,000/month.
8. Career/education wise, I’m working on my master’s in engineering and my employer is paying for it. My girlfriend has about a year left of school for her medical imaging degree, which her hospital is also paying for.

So neither of us expects to take on additional student loan debt for our current degrees. We’re in Indiana, and decent 3 bed/2 bath houses in our area are around $225k-$275k, with ~$250k being pretty typical. We’ve been considering saving until we have around $50k+ available for the house. Depending on interest rates, taxes, insurance, etc., it looks like our total monthly housing payment could potentially be similar to or even less than what we’re currently paying in rent. My biggest concern is not becoming house poor or dumping our entire savings into the down payment. We both grew up pretty poor, so having financial stability and eventually breaking that cycle for our future family is extremely important to us.

Usually I would ask my dad, but he passed away from cancer last month, so I’m hoping that this beautiful forum would have some good minded people to guide us. My father was also a father figure in my girlfriend’s life as she didn’t grow up with a dad.

Would you consider us financially ready to buy a ~$250k house? How much would you put down in our situation, and how much cash would you want left over after the down payment and closing costs?
Would you continue renting for another year and build up savings, or start seriously looking at buying now?