Sounds like either thermostat gasket or housing is leaking, hose leaking, or clamp not tightened causing the leak.
If hose is real swollen at end near the thermostat, the clamp is most likely cutting into it at this point- you would need to replace hose.
If you are unable to identify leak on your own, you would need to have a cooling systme pressure test done which positively identifies leak. Most shops charge $30-35 for this test. Worth the price to make sure you are replacing the right part and not guessing.
Make sure you check it between every place you turn car off at. I would not go too long. A slow leak can suddenly become a severe leak which could cause vehicle to overheat and lead to extermely expensive repairs.