Phew you've got quite the conundrum on your hands!
I'm a huge advocate of the HELOC, or Home Equity Line of Credit. It can usually be done at your local credit union and is usually far less expensive than a cash out refinance. This would allow you to take a HELOC out on all three homes and hopefully utilize that equity to get them to rentable condition. This would allow you to pay yourself back on the HELOC from the generated income allowing you to increase the values of the current homes without putting your personal assets/liquid cash at risk.
I hope this helps and that's 100% the route I would take if I were in your shoes. Depending on where they are, you may consider STRs and generate even quicker cash flow. You'd want them in some pretty tip-top shape if you were going to take that route though. I'm confident that if you took that route you'd be able to pay yourself back in just about half the time though. Feel free to reach out if you'd like to discuss further. I'd be happy to help. I live in the 15th and Union area myself, so I'm very familiar with the area.