I want to build a drone that can carry around 7lbs. Is it possible to make a drone that can carry that much weight using electric motors without it being over /$500?
Yes it is.
You wanna carry roughly 3.5 kg. Let's make the drone with payload 10kg heavy. For a quad rotor, with TWR of 2, I can get the 4 motors for 1200\$ (public price from T-motor U8 Pro), I don't know the price of the propellers, let's say 400\$. The whole makes 1kg. Adding 3 kg for the frame (I roughly scale down from a 25kg drone) and 2500\$ (here, I have no idea how much it would cost), we've got 2.5kg for Lithium Cells, it's roughly a 10Ah 12S battery, for 500\$. We may add 200\$ for electronics and sensors (and roughly no weight).
You've got your drone for less than 5000\$, flying 20 minutes. It excludes the cost of the time you'll spend in development, and also the cost of the prototypes you'll need to correct the approximations I made, unless you accept performances far from this first approximation.
If you plan to design your own propeller, you'll probably need 100000$ for CFD tools, prototypes, and measurement tools. If you want a cheap prototype, use off-the-shelf components. If you plan to build thousands of drones, then the prices above are irrelevant, and you should consider some serious R&D.