My apartment in Florence requires a damage deposit in check or Travelers check

Easier for you to acquire euros. Are you sure they won't accept cash?

Since it is a refundable deposit and likely will not be cashed (They will simply hold the check in the event there are issues) then I would simply ask if a Travelers Check, or a personal check, in Dollars, would be OK.

Getting a Travelers Check may have some costs, there are none if I get them through my credit union (though I have not bought one in a long time). You can get TC's in euros, but that usually incurs more costs, paying for a conversion and fee to euros, and maybe back if you do not use it. Cash would be another option, but some risk, plus you might wind up with a bunch of euros you won't use.

you can get a international money order or bank draft at many banks. You have to ask someone in authority - tellers don't always know what you're talking about. For example, I recently bought one denominated in Pounds Sterling at a local branch of USBank.