What sort of things did he pay that were not 'related to the administration of the estate'? Sounds a bit odd - did he think they were or was he just helping himself?
Issuing the final account has no bearing on whether the debts are payable from the estate or not - that's governed by the statute of limitations. There are circumstances where the executor is responsible for settling the debts himself, but still has the right to ask the beneficiaries to pay up.
How much will the claim be? The small claims court is the obvious way forward if the amount is within their limit..