I don’t think you should buy another house if you are in a poor cash position. Even if a lender can do the deal you have detailed, it has a high probability of failure on your part. If the renter doesn’t their rent you are in a real financial bind. If the property needs work (water heater, A/C goes out, etc…) you have a true financial disaster on your hands.
Fix your house and have 6 months expenses in cash reserves and THEN you will be able to move forward.
I am looking for a lender who will work with me on a purchase of a new home (either brand new in up and coming area with good appreciation) or newer home going in with equity built into the deal. I have an existing home that I would really like to keep as a rental property but it needs about $5000 in cosmetic work to get it ready. I have little to no extra cash for out of pocket expenses or to pay two mortgages at once. Are there lenders who will work with me to finance the new home purchase and roll in some extra cash to cover the gap to get the existing home rented?