Why is there heavy oscillation in my capacitor discharge circuit and is there anyway to fix it?
What you have here is a classic LC tank circuit.  This is basically guaranteed to oscillate.  Changing the sizes of the inductor and capacitor will change the frequency and adding a resistor will damp the oscillation if it's large enough.  It might help if you hang a load off of that transformer so the cap has something to discharge into.  Right now, it seems like it discharges into the transformer coil, which then charges the cap right back up in the other direction due to the inductance.  If you put a load on the other side of the transformer, more of the energy will end up in the load instead of back in the capacitor.