Is it safe to have both accounts at the same bank or credit union? All our accounts at one place seem to be linked to a certain degree

Would a new account at the same bank work or are we looking at a new bank all together? Would be easier to transfer money into the old account for bill etc that are set up

Only going back to finish following creators on other platforms at this point. Putting the time into learning Spanish and possibly Arabic to be able to talk to coworkers easier if they don't get deported