I live in Texas and my water well and tank are outside.

Last week, I struggled to change my pressure valve. The previous person who repaired the well did not use teflon tape so it was extremely hard to get off. While trying to loosen the old valve the galvanized tank wiggled/moved.

The valve was finally replaced when I enrolled my neighbors husband and his huge tools. One the valve was loosened it took 5 minutes to replace without any leaks; this time with teflon tape. However, now there is a trickle of water coming out of the bottom of the galvanized tank. I noticed the tank looks like it is leaning. I am going out there today to empty and try to level.

My concern is that I ruptured the bladder when I shook the tank and now need to replace my whole galvanized tank. You folks who have had more experience in this area, what are your thoughts?

Update, emptied tank. Looks like the wiggling jarred the connection pipes from the well because there are a few leeks from those old galvanized pipes. But there is still a trickle out of bottom tank.

First, there is not a pressure valve on a well tank. There is a pressure switch and there is a pressure gauge. If you have a galvanized tank it is not likely a bladder tank. As far as I know all bladder tanks are painted.

Yes, if you have old galvanized piping you could cause a leak at a joint by moving the tank around. Galvanized pipe last a long time. Corrosion of galvanized pipe is almost always at the joints. That is because the galvanizing is remove when cutting the threads. Moving the tank around could cause leaking at old corroded joints. If you have old galvanized piping I suggest that you remove as much as possible, install a threaded PVC adapter to the galvanized pipe and at the tank and then repipe with a Schedule 40 PVC pipe. Repiping with PVC is cheap and easy. Teflon tape is a great thing but I prefer pipe dope (pipe thread sealant) for such situations. Not really all that important that the tank sit level.

You said

But there is still a trickle out of bottom tank.

If that means that there is a leak at a joint, repiing with PVC will fix that. Repiping with PVC is easy and cheap.
If it means that the leak is from the tank itself, the tank will have to be replaced.

You do know that pressure tanks (galvanized), not painted bladder tanks, become water logged and have to be periodically recharged with air.

Process is simple. Turn off pump, open a faucet some where to let air in and then open drain valve on tank. If there is no drain valve on the tank, disconnect the well pipe. There will be a union somewhere in the piping from the well where you can do that. When tank is completely drained, close everything and turn pump back on.