It must be a "line of credit" where I can make draws, and not a home equity loan. 

If there is enough equity within the properties based on their appraised value, you may be able to do a packaged loan for these two properties and then have enough equity for a down payment on another property.

I know loans can take 30 or more days to close.

They are working on a refinance of a private loan I am the lender on and I am just looking to see if they are the real deal or not.

Loan Options: Utilizing an FHA loan is helpful since it requires little down.

Otherwise, how will the portfolio's performance improve? 

We are open to getting a home renovation loan and finding something cheaper and renovating, but we still can't get into the prime areas for the budget.We have been advised to make sure to choose an area insulated with good schools and parks but that's just not possible with our budget unless we go to the suburbs, which we are not willing to do.

Make sure that your lender/loan officer is willing to get on a call with your lawyer to help out with any discussion/details to dial the agreement in as much as possible.

So, that is you refinance your cash-flow will improve quite a bit. 4.

I have a good credit score, but I'm concerned that given I'm going to have very low provable income for the next 2-3 years and not be able to secure a loan especially with a low down payment.