I am sorry for the delay. I usually don't do much on the site until the afternoons.
Tomorrow, call the family friend and read him the letter that you got. It will explain what you're entitled to do to in order to refute the allegation of fraud. There should be some administrative appeal process mentioned, but you don't want to make any statements or attend any hearings without letting a lawyer know what you plan on saying or having him attend a hearing with you. The government makes mistakes just like anyone else, and if you did nothing wrong, you may be plenty inconvenienced before all is said and done, but it should come out in your favor.
As you have not yet been charged, you're worrying prematurely. In order to search your house the police would need a warrant for your arrest. In order to get one, they would need to convince a judge that criminal activity may be going on and that evidence of it would likely turn up in your residence. If the judge believes that what the police want is reasonable, having heard what the police tell him about the situation, then there is probable cause
for the warrant. It's been my experience that my clients charged with various types of benefits fraud have not been subjected to search. But I cannot guarantee it won't happen.
You don't have to prove anything at all A defendant on a criminal case can (and should) remain absolutely silent and let her lawyer make any statements. It's the state that has to prove its case, and they have to do it beyond a reasonable doubt. Never a good idea to help them out by making what you think are innocent statements but which can frequently prove damaging in ways that non-lawyers wouldn't see.
Frankly, my ex and I divorced double-digit years ago and I still have some things he's never bothered to pick up. It's neither illegal nor unusual to still have some of your husband's property. I cannot say you're not in potential trouble, of course. But I think you'll likely feel better