@DebtAfterDeath - That all sounds good. It is a difficult time for you all.
I can't believe how many people have given you totally incorrect advice and seem to think they know it all. Some advice is so way off as to be fraudulent behaviour.
When contacting people to tell them about the death, your SIL and DH don't need to mention that they aren't sure if there will be enough money in his estate initially, as you may find that there is indeed enough in the Estate once all income and bills has been accounted for.
They simply need to advise of the death and find out if there is a debt to be paid or money to come back. My late Mother got a tax rebate, and a refund on her Council Tax, plus, the council tax on the property was suspended from the date of her death until her rented property went back to the owner. A few other things too that I expected to have to make further payments on which were either cancelled immediately or I received small rebates from. Companies understand that it takes time to settle an estate. Many of them have 'bereavement departments' too who are easier to deal with and tend to be more clued up on what happens to any debt
Once you KNOW the exact position, then is the time to either pay all outstanding debts from his Estate or pay the priority ones first as listed and then contact the rest who still want cash. It sounds daunting - it probably won't be as bad as it sounds
My Mother had less money than your FIL in her accounts and initially I thought I was going to be about £2-3k short to pay all that needed to be paid. In the end there was around £150 left in credit!
Remember, reasonable expenses incurred for settling the Estate can be taken by whoever is dealing with settling it: stamps, petrol money, cost of death certificate(s) etc etc
I dealt with both my parents passing in Scotland, so rules are not entirely the same, but most of them are
Condolences to you, your DH and your SIL and FIL's and others who knew him