Hello Gerardo,
Happy to answer your questions. What many people miss is that a quote is a quote and nothing more. Banks/lenders quote all day and some may not be honest because they want to hook you in and make you a customer.
Nothing is official until you get an official rate and terms sheet/loan estimate, as wall as talk to someone with the bank/lender who goes over their loan pricing and what your options could be. Also, even when a lender can lend, if you fail underwriting you may not be qualified for the loan.
So ask them for an official loan estimate.
You also have mentioned that banks are become more selective. Absolutely true. With interest rates quite high now, lending is risky to banks so they will be more selective. I would also say if you want to increase your chances of getting some financing options despite all these current market situations I would recommend you call a local mortgage broker near you.
Mortgage brokers unlike banks are able to shop multiple banks/lenders to find you options. Usually, the more banks/lenders they can call, the better your option of finding someone who will approve your financing request as well as finding the best financing offer(interest rates, terms, etc) despite the current market conditions.
I hope some of my answers were helpful to you. Best of luck!