Why does my GFCI receptacle trip when anything is plugged in?
It certainly seems like you have a ground fault somewhere. Is it possible there is water damage anywhere inside the garage?

Since you've already replaced the GFCI, I think it's safe to assume it's working properly and is tripping as it's designed to. That means somewhere current from the hot or neutral wires is leaking to the ground wire or somewhere else. (I'm guessing it's a neutral -> ground short, since it only happens when you try to use an appliance. A hot -> ground short should trip the GFCI as soon as the power comes on.) 

Here are the things I would try:


Does the tripping happen at the GFCI, or just the downstream outlets? If it's easy to tell which order the receptacles are wired in, start by plugging something in to the first one downstream of the GFCI. Does that trip it? If so, the short is probably between that outlet and the GFCI. If it doesn't trip, move to the next outlet and see if that trips, etc.
Take the cover off all the outlets and inspect them. Look for any signs of moisture, or a loose wire nut, or a conductor touching the side of the box.
If you don't see anything suspicious, start pulling the outlets out and rewiring them. Disconnect them all and start adding them back one by one.
If you still haven't found the issue, it's possible the wiring inside the wall got damaged somehow. Possibly a combination of mice chewing on the wires and moisture from the storm? If you've reconnected the wiring to the outlets one by one, starting with the closets, you should at least be able to tell which section of wall the defect is likely to be in.


Other notes:


I'm not sure if the continuity tester will help much, the problem is a short not an open circuit. Especially if the problem is caused by a neutral / ground short, those are always going to test as continuous since they connect at the main breaker box.
I wouldn't worry too much about the voltage readings you got with the GFCI tripped, open circuits can sometimes have small but non-zero voltage readings.