That is correct; their annual HSA contribution limit is $7,300, because they are both considered to be under a Family plan, never mind that one spouse also had a Self-only plan.
So, yes, if they contributed to one HSA, then they will need to contact the HSA custodian and ask to "Withdraw excess contributions" (this keeps the paperwork at the custodian correct). For many custodians, there will be a form on their website to do this; otherwise just call them. If they spread their contributions over two HSAs, then they can withdraw the excess from either HSA so line as no HSA got more than $7,300.
"Does a regular claim for a medical expense count, or do you somehow need to contact the bank and request money?" No, reimbursing medical expenses does not count - you must contact the HSA custodian as noted above and ask for a return of the contributions. Be sure to use the phrase "withdrawal of excess contributions" so that the custodian does not mark this internally as a distribution which would be penalized at 20%.
TurboTax will add the amount of the excess to their Other Income, so they will pay tax on it. But they will receive a check from the HSA custodian for the amount of the excess, so they will be able to cover the additional tax.
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