You have several options in your situation:
1. Do not make the repairs. (Assuming you can legally sell without
doing them) Have you done the math to see if you can get your repair
money back out when you sell? This is the safest option for you and
your money if it is feasible.
2. Refinance. If you have a substantial amount of equity you could
always refinance your home. Yes, it will cost you some money, but as
a last resort there is no reason you couldn't do it. Yes, interest
rates are probably higher now than what you have, but if you are
selling it soon, who cares?
3. Home Equity Loan. I'm not sure where you got the information on
your ineligibility for a home equity loan, but I know several people
that have taken a loan on the equity in their homes to purchase
another house. All the while trying to sell the house with the equity
loan on it. I consider this your second best option.
"Home equity loan - Many people take out a home equity loan against
their existing house and use that money to put a down payment on the
new one. If you do it right, you should be able to make a down payment
and have funds to make the mortgage payment for a few months while you
sell the other house."
I hope that this gives you somewhere to look. Please let me know if
you are unsatisfied with this answer. My father was a senior loan
officer at a bank for many years and I can get his advice if you need