If it holds air for a week then I would check it again. Bladder tanks don’t normally act as you described I would be wery of that tank. When was the last time you disinfected it? If your pump kicks on at 30 psi you should put 32 psi in the tank for operating purposes.
The only way an expansion tank can drain is for air behind the bladder to push it out. If your tank does not drain completely than it does not have enough air in it to displace the water inside.
To drain an expansion tank
First you need to drain the tank as far as it can go by itself. Next using a bicycle tire pump fill up the tank only until the water begins to flow out. Do not over pump the tank or you may ruin the bladder. Keep doing this until you are sure all of the air is out of the tank. Now to obtain the propper air pressure this can only be checked with the tank empty. Most well tanks use 30-35 PSI empty. Now it would be a good time to disinfect the tank with some bleach. Use an empty filter sump housing to introduce about a 1/4 cup of bleach into the tank you will need some valves to do this. Now rinse the tank by filling it and draining it sevrall times.