Also...with the major traveller cheque people...you can call a helpline and have them cancelled. (I'm pretty sure...it's been a few years since I used them. For me they are a thing of the past. We've used our debit cards since the day they came out...makes traveller's cheques obsolete.)
Is this story set in the current century? Because like everyone else has said, traveler's checks are obsolete. I was a hotel manager from 2000 to 2004 and in that time I maybe saw two or three. (And, I was the person who counted deposits, so I saw every one that the hotel took in.) It's all credit cards and debit cards now. I guess a character who couldn't get a credit or debit card might use them, but they'd be more likely to use cash or stay home.
Today you can use them to pay for your room, but most hotels will only cash $50 to $100 in travellers checks for you.
I would also hazard a guess that today, most restaurants outside resort areas won't take them. I think you'd have to go to a bank or a hotel to use them, and at a hotel we would ask for a photo ID. And, both signatures on the checks better match the signature on the ID.
In the 80s and earlier, American Express had a system set up so that travelers could get their lost or stolen travelers checks replaced at hotels. The hotel had a supply of new, unsigned travelers checks sitting in the hotel safe. Joe Blow would call Amex and report his travellers checks stolen. American Express would call the hotel manager and say, "Mr. Joe Blow is going to come in. He needs $500 in replacement checks." He would show us a photo ID and we would give him the traveler's checks. (I think $500 was the maximum, but it worked even if he was not a guest at our hotel. It worked even at 2 am on Sunday.)
Also, I believe that if you report them stolen and someone does cash them, the issuing company -- not the individual -- is out the money. So, it's Amex's loss, not the customer's. (You need to know the check numbers to report them stolen, and you're supposed to make copies of the checks after you sign them, although few people do.) If they are cashed before they are reported stolen, I believe that's the customer's loss.