Direct Deposit to new checkings account?

So, my husband and I both receive direct deposit to our checking accounts with Wells Fargo, a bank we will be trying to settle our credit cards with.

Do you recommend that we open a checkings account with another bank so that Wells Fargo does not have a record of how much money we are receiving? or does that not really matter?

Yes , change banks asap. You have whats known as cross collateralization which basically means if you stop paying on a credit card and you bank with them they can just take the money from your checking or savings.