Hi @Abigail Glover, I see a few things here.
1. Since you do not want to do a cash-out, you can look into a HELOC. This would not change your current interest rate, but it would allow you to access your equity. You can use this money to fund the down payment for the flip.
2. The issue with buying at auction is that you typically need a cashier's check or the funds available to purchase the home within 24 hours of the sale. Traditional lenders cannot meet this requirement so you would need to use a hard money lender. The problem you run into there is that they will only lend on experience.
You have two options with that in mind.
1. If you are looking for properties that will cost $200k to buy and rehab, you would have enough funds to buy it cash at auction after closing on your HELOCs. You could then self-fund the rehab and pay off the HELOC in a few months once the property sells.
2. You can get connected with wholesalers or off-market sellers to find people that do not mind a 30-day closing. This will give you time to inspect the property and ensure there are no hidden expenses as well as obtain financing. You can get a fix and flip loan with 6 or 12-month interest-only payments to keep your carrying costs low while covering most of the purchase price and rehab.
Hope this helps! Let me know if I can be of any assistance.