I don't doubt you have leaking caps, but I don't see anything in the pictures... The globby white stuff is normal hot glue on the analog board I believe? From what I've seen caps leak a clear liquid that's difficult to even see sometimes. I think the capacitors that are leaking are probably on the logic board and not the analog board.

Soldering this kind of thing is not difficult. Find some old broken electronic stuff with big capacitors (i.e. an old TV or VCR, etc.) and use a soldering iron to remove some components. You'll get the hang of it. Then you can try putting them back in and repeating the process with smaller components. Be careful not to burn yourself on the little guys, they get hot quite quickly even on the opposite side of the board.

You will need an iron, some solder, and maybe one of those solder vacuum things. You push the shaft down to load it up, then press the button to fire, and it sucks up hot solder. These things can all be gotten at RadioShack.