I don't know what a "short body die" is. Do you mean small base dies? If so, no you shouldn't need them for the handi. If you cannot chamber a case fired in your own chamber after sizing it you have a major die problem but not one that requires a different type die to fix.
Any case fired in a chamber should go back in that same chamber with no sizing. It just came out of it, it should go back.
If it won't then you've altered the case in a way to change some dimension that prevents it. Case might have grown in length. They do during the sizing operation and if too long must be trimmed. If you have your bullet seater die set incorrectly you can cause a buldge that will prevent it from chambering again.
I suggest you first try the case back in chamber after firing. Will it fit? It should. Next size it. Now try it again. Will it fit? It should, if it don't the sizer is the problem. Find out why. If it fits now and not after you seat the bullet then you've done something in the bullet seating phase that's altered the case or the bullet is seated too long. Work it out step by step.
