Yes, you can, although you may find it easier to use the remaining funds in your older account first. As long as you remained HSA eligible the entire time, the new account is treated as having begun when the old account did. If there was a gap in time (lapse of qualifying HDHP insurance, for example), then the new bank account could not be used for the expenses. It would be treated as a new HSA account, and you cannot use a new HSA account to pay on already existing expenses. But your details state that you had qualifying insurance the entire time. No issues in using the new account.
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