I’m urgently looking for a bank that doesn’t use ChexSystems when opening an account. 
If anyone knows of any reliable banks or credit unions that offer this, please share your recommendations and experiences.
Any help would be greatly appreciated! 
I would start with local credit unions and second-chance checking accounts, since many of them do not rely on ChexSystems in the same way as large banks. Ask directly whether ChexSystems is checked, and what happens if there is an old record, before you submit a full application. Also compare monthly fees, minimum balance rules, and whether direct deposit can waive those fees. It helps to bring a valid photo ID and proof of address, and to keep a small opening deposit ready. If one application is declined, ask for the reason in writing so you can fix the specific issue before trying another account.
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