I truly only need 3.5% down for a home purchase (primary residence), with an FHA loan?

I’m very new to home buying obviously and this will be my first home. I’m in the infancy of looking and researching.

I have a 780 credit score, my only debt is $7,000 loan for my MBA (payment is less than $100 each month) and I’m figuring in this current market, I’m hoping to get in a house around $250-$300,000.

Going with the high end of the scale: 3.5% of $300,000 = $10,500 + closing costs and then I’d still have to pay PMI each month.

I’m going to need to increase my monthly income with a new job and I’m actively looking, but I have enough in savings for the down payment (3.5%) and closing costs, so once I increase my income, could I maybe get approved if I stay this close to debt free and no car loan?