I would never use only one bank for all of my accounts. If your bank ever shuts you down, you don't want to be scrambling to start 10 new bank accounts.
I'm not sure why the state of Florida is "burned" for you, but if you are losing bank accounts, something is not right. To lose accounts at one bank could happen to anyone. But if all banks are shutting you down, you might want to reconsider your business practices.
Having said that, there are many banks that work for stealth. Especially with LLC's. If you are setting up business accounts and the LLC names match the account names, then you should not have any issues with mismatched names.
Another consideration is ChexSystems. If you are in their database and have left banks "holding the bag" so to speak, it will be more difficult to find banks to work with you. For those in ChexSystems, you might want to google "second chance banks" or "no ChexSystems". Credit Unions can also be an option.
WoodForest is a good second chance bank that works for stealth.
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