Yes, in that budget get a safety razor or if you are really set on shaving with more exposed blade, a shavette type of razors could be found quite cheaply.
You want to stop by walmart/grocery store and pick a shaving soap for $1-$1.50 and a brush for $5 (your budget doesn't allow anything else) - then spend half an hour making lather over and over until you figure it out. Proper beard/skin preparation is non-negotiable for a good shave regardless of how you cut the hair.
I think you'd be better off getting the tweezerman brush right now and get the razor in another few weeks when you have set aside a bit more money. Though if you don't have anything else to shave with right now may as well get a razor and DE blades instead of another pack of bic's.
(I suspect you have problem with them because you do not prepare your face well and you're using them for longer than the 1-2 shaves that they're good for - unless you strop the edge that's how much you'll get out of it regardless of whether it's a cartdridge, de/injector type blade, or straight razor.)