Generally speaking, Fidelity is top notch in both customer service and security. Every once in a while, there is some glitch, although the OP's issue seems excessive. Never heard that one before.
Probably not a bad idea to keep a month's worth of expenses in another institution if you can, just in case.
If you don't have an assigned financial advisor at Fidelity, you can go to the LSL office on Monday and try to resolve in person.
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