This is NOT something that can really be diagnised for sure without me actually being there and doing tome testing by driving it and so forth.
But most likely what you are going to find out is that the ENTIRE transmission is going to have to be replaced.
These usually are not something that we recommend that they be rebuilt.
They usually do not work very well if they are taken apart and the cost of rebuilding them usually are not too far from the cost of replacing them..
But of you want to try to check this out, I am including a complete parts breakdown of the transmission for you.http://www.partstree.com/parts/?lc=mtd&mn=13AR606P730+%282005%29&dn=07305000030
The part number for the complete transmission is 753-05851
The actual problem is most likely in the two gears #15 in the diagramThe clutch collar #16 in the diagramThe Detent shaft #20 in the diagramAnd probably the detent ball and spring #21 & #22There is also a good chance thet the shaft #17 may also be worn.But the gears and the clutch collar are almost always bad along with the detent shaft.
Actually if you really want my opinion, I wouldn't even bother fixing the tractor at all, I would buy a new tractor.....
But if you are asking if I would fix the transmission or replace the transmission, I would Replace the transmission.
Usually the factory doesn't sell to customers, but you might check with them.
Online usually will give you a better price. Check www.sears.com/parts They usually have a decent prices.