If your Handi was manufactured in 93/94, the second letter in your serial number would be: H for 1993, or I for 1994.
H&R won't fit a barrel for you. They only fit barrels to rifles made 2000 (prefix P - 2000) and after. This doesn't mean you can't another barrel yourself. You can probably find an extra barrel in the caliber you want in the classifieds.