As a private lender, I would do the following:
1. Transfer your mother's house into an LLC and rent out (likely as an LTR)
2. Take a DSCR cash-out refinance on your mother's house with a hard money/private lender. They will not use your DTI to qualify you, only the DSCR (essentially monthly effective gross income minus insurance, taxes, and HOA fees), LTV, and your FICO credit score.
3. With the cash-out proceeds, you cannot pay off current debt, but you can invest in more real estate offering a higher rate of return than the debt (ex. 15% return on a house vs 11% rate on debt, the investment has less opportunity cost. I would use your household income of $200k to address the debt.
3a. I would use the cash-out proceeds to put down payments on up to 3 additional rentals with a private lender DSCR product.
4. Use a conventional owner-occupied home loan to purchase your personal home. I would look at an FHA-type loan with 3.5% down plus PMI or a 5% down owner-occupied option. This will give you the most leverage and you will allow you to add even more properties to your portfolio, increasing return and decreasing the risk.