@David G West Hey David! Congrats on starting your journey! There are a couple things to note in your questions.
First, if you start working 1099 in June, you will not be able to qualify for a conventional or FHA loan. As a 1099 independent contractor, you are seen as "Self-Employed" and you will need a 2-year history. There may be SOME flexibility to allow for only a 12-month history depending on your specific scenario, but if you just graduated that is unlikely.
Second, I would definitely recommend saving more for the down payment, but whether you pay off the student loans or not would depend on your income level. Even if they are in forbearance, they will still impact your debt-to-income ratio since you are looking at a 30-year mortgage and they aren't going to be in forbearance for the whole 30 years. If your income level supports the added "payment" they would calculate (1% of the balance, so ~$300/mo) then I would say save more for the down payment and don't bother paying them off.
Hope that sheds some light! You're off to a great start, nice work!