Can you sell your home to a third party and then purchase it from them in order to refinance?

Well, in theory you could, but practically speaking probably not. Purchases require stricter underwriting guidelines than refinances, are more costly, and require down payments. FHA loans, which have the lowest down payment at 3.5%, also require an unfront mortgage insurance premium cost of 1.75%. Bottom line, if you are unable to refinance now (which seems to be your goal), I see no circumstances under which you’d be able to buy the home back from a third party buyer/seller. In addition, this party would likely be someone you at least know (and may be related to) which would make both the sale and the purchase “non arms length” transactions, entailing even more scrutiny. Why are you unable to refinance in your own name now?