Go back and jumper around the relay like you did before and let the compressor run long enough to see if it's going to cool. You won't hurt anything by doing that as long as the blower fan is running. BTW, the condenser is not powered, it has air blown through it at the same time as the evaporator with a single fan. If the unit blows cold air with everything running, then you have a bad control board.
Hang on to the caps you replaced. It never hurts to have spares. Everyone jumps on the start capacitors because they know nothing about AC units, but they read a "my brother-in-law had that problem" post somewhere on the Internet. In reality, capacitors just don't fail very often and when they do, it's usually visible like the can bulging or it's leaking electrolyte. In over 15 years, I have only had one or two that looked good and checked bad.