Definitely need your circuit to be sure what's going on.
Where did you get you MOSFETs? There are a lot of counterfeits about, and anyway you need logic-level if you want to drive the gate direct from 5V. Otherwise you'll have to boost the signal to 12V as this is what the IRF520 is designed to work with.
A large power MOSFET will have some drain-source leakage current, measured in microamps, this is normal.