It really depends on what other debt you have (i.e. car payments, minimum credit card payments, student loans, other real estate, etc.) and how much the new house payment will be.

Generally for a conventional loan your total debt cannot exceed 45% of your gross/before tax income.

On the other hand, VA does not have a "debt to income" ratio limit, but generally lenders want to see no more than 55%. If you have high credit scores and good cash assets, the 55% limit can be exceeded. Not to bore you with too many underwriting details, pretty much if you have good credit, some cash in the bank, and fulfill the VA residual income requirements (income left over after paying off all monthly bills), you are good to go.

I understand your concern about hard credit pulls. However, one hard mortgage credit pull now will not lower your scores much if you already have good credit...Maybe a point of two. I have helped many buyers in your exact circumstances, and if their scores are already good, often there is little to no change when we re-run credit. Credit reports are good for 120 days.

Feel free to PM me for more info. I am a mortgage loan officer who specializes in VA financing. Most of my clients are pilots.