Attack strategy for $60,000 of student debt

I'm currently on SAVE and in forbearance, but interest is accruing and I am finally in a position where I can begin to pay down student loans. I want to use my time left before I am forced onto another repayment plan to pay down as much as I can on targeted loans and hopefully reduce my monthly payments on whatever I end up switching to.

I have 10 student loans from one undergrad and one graduate degree. I wanted to start with my highest interest loan at 7.08 percent, which had a principal of $11,586. With a recent payment from my Americorps education award, I was able to cut through the interest on it and get it down to $10,566 and I am energized to start paying it down aggressively now that a 4-digit number is on the horizon.

HOWEVER, I am very worried about my $20,500 loan at 6.08 percent interest, which in the meantime has racked up $1,400 in interest.

All the other loans are below $6,000 and are my lowest priority that I figure I can deal with later down the line.

I have at most $500 a month left after living expenses to put toward loan payments. How would you target your money to handle this situation?

I do also have about $15,000 in savings, but I really wouldn't like to touch any of it for loan repayment if possible as I'd like to save for a home someday and it would take a very long time to start over.

I did not get good financial advice (or really much advice at all) when taking out these loans and I feel a bit embarrassed and behind the ball when it comes to repayment. Thanks for any advice you have.