@Casey Martin The comment above about your Mom selling tax free is definitely something I'd take into consideration if nothing has changed hands yet.
From a mortgage perspective, it sounds like you might have enough room in your DTI to make this work if you don't have much other debt besides the student loan and the mortgages, although I have no idea what your current interest rate is, or your property taxes / insurance, or whether all your income can be counted. For that, you'd definitely need to talk to a loan officer licensed in your state to review your financials.
Here's what I'd suggest:
1. Cash out refinance on the gifted home - you will need to be on title for 6+ months to qualify for a conventional cash out refinance. You can borrow up to 75% of the value for a cash out refi, and can use 75% of the rent to qualify. You will typically need to show the lease and first month rent / security deposit to use this income before it's reported on your tax return.
When refinancing, any debt that will be paid off with the loan proceeds can typically be omitted from your debt to income ratio.
2. If you can show that your Mom has insufficient income to qualify for a new mortgage, you may be able to take advantage of Fannie Mae guidelines that allow you to buy a home for your parent(s) as a primary residence. This will allow you to put a minimum of 5% down and you would get the better owner occupied loan interest rates. Since you would not be buying this as an investment property, you cannot use any rental income to help qualify for the purchase loan, and you would need to qualify this additional mortgage payment (principal, interest, taxes, insurance, HOA). If you buy as an investment property, rates will be higher and you will need a larger down payment.
So, first you need to be on title to your Mom's home for 6 months. Then it needs to be leased, then you can do a cash out refinance. Once you have paid off your debt, have the home leased and you have your down payment funds, you can buy the new home for your Mom to live in.
Obviously before taking the first step, consult with a tax advisor to see if they recommend a better option, and also consult with a loan officer to plan out the scenario to make sure you will qualify for each loan.