What are some of the average mortgage rates in Houston?

My fiancé and I are soon to be applying for pre-approvals for a mortgage next month and I was just trying to gauge what I should be aiming for interest rate wise. We are using our realtors "connections" but would be open to applying at other lenders as well. A couple months ago my coworker, who is also looking for a home, said her lender told her that the lowest possible rate you can get is 4.5%, however just yesterday she told me that her lender told her the lowest you can get is 5.5%.

Also is it true that after you apply for your first mortgage loan, and it appears/effects your credit, you have a 14 day period where you can send out more applications, that wont EACH effect your credit score, in order to shop around for the best rates?

Just a bit of information:

My fiance and I have a DTI ratio around 9% (depending on if they include future loan payments. Currently dont have to pay student loans and will not be paying them at the time of application, however the student loan payment pause ends in September so I will resume paying them at that time. Without student loan payments our DTI ratio is 7%)

Her credit score is 810+ and mine is 790+ and we make about $112,500 combined annually and lastly we are looking in the NW area of the outer suburbs like Cypress, Katy, Tomball. Our budget for the home price is $285k.

Would anyone be willing to share their mortgage rate that they recently received in the same area if possible or share any idea of what interest rate we would receive when we apply?