Few are grafted or cloned , so by and large there aren't really varities , just types ... I've grown them in quanity ( Bay Islands Honduras ) and most turn out as you have discribed yours , with just an occasional large fruited specium . ( I've grafted these onto new seedlings with some sucess .
You can adjust height when young , by scoring the young seedlings ,to force it to remain short or sometime branch. It's my belief that the fruit will get smaller for you now and you we'll need to replant...Their fruitful lives are wonderfully fast , but sadly short....richard

I don't know what type of papaya you have but our trees grow huge - we have a couple at 20 feet. When the get so high that harvesting becomes a problem we cut down the tree and PUT A BUCKET OR PLASTIC COVERING OVER THE STUMP (WATER GETTING INTO THE CUT STUMP WILL ROT IT IN NO TIME) The papaya then sends up side branches. The first time we did this was frighening - but it worked for us! I believe there are some cultivars that stay shorter. Good luck! You may want to check out the Hawaii forum, we have lots of great gardeners who grow papaya there. Jenny.

Papaya just keep getting taller and taller. One down the street is easily 25' tall and 6 or 8" in diameter, with huge Papayas. Well they probably do die, I mean they wont get 100' tall or anything. The variety you have will only produce papayas of that size. I agree, try the hatchet/bucket trick, but (if that fails) they are really easy to grow from seed. I have 3 or 4 varieties. Wilamano(fruits at 3'), Carib. Red (huge 4lb papayas), Red Lady (fruits at 4', and self polinating), and FL wild naturalized Papaya.

Check out Aloha Seed Co:
http://www.alohaseed.com/cgi-loc/webcart.cgi?RET=125&RCATS=Papaya+Seed&FIRST_TIME=Yes&START=1&OCATS=Specials&DOSEARCH=YES&CODE=150

Or BananaPlants.net for 2kg papayas:
http://www.bananaplants.net/papayas.html

I have had great luck starting them from seeds from Food Store Papayas.