Check youtube for some videos on rebuilding the Peerless trans - doesn't look too difficult if all you need to do is change a gear or two, or maybe put in new shift keys.

Before you do anything else remove the mower deck and carefully inspect the drive belt (from engine pulley to trans pulley), all the idler pulleys and belt tensioner mechanism.

Your problem may be that there is no slack on the belt at a stop making it difficult to put the trans in gear. Stop the tractor, set the brake, and verify that there is slack in the drive belt.

There's aso a possibility that the previous owner replaced the drive belt with an incorrect one that is too short. Check the drive belt for part number and verify that it is the correct real Sears belt for that model.

Manual LT/GT transmissions are not like car transmissions. You must stop and choose the gear you want then go.

Thanks for the input so far, keep it coming. I really appreciate it.

The deck is off and inspected. In fact the symptoms I report are all with the deck off. The previous owner had taken the transaxle off and opened it and reported no visible signs of wear that he could see, no shavings, broken sprockets. He did not delve in any further than taking a look, he said it looked pretty complicated and he had found a great deal on another mower so saw no need to pursue. He sealed and put back together and mounted it.

If it is a rounded shift key he probably needed to remove things to check that I would guess. From all my research to this point and talking to several mechanics, that is there guess without looking.

I personally have never ventured into any thing like a transaxle, but doesn't mean I won't, just pretty apprehensive about it, and I need to be mowing yesterday. I am sure I am still weeks away from a fix.

Always start at square one and look for the simplest things first.

Pulled the transaxle apart and there is little wear and tear on it. Looked very good. You can see a little minute wear on the shift keys, and from what I have read, that is all it takes to make those not perform perfectly. Dug around in the grease with a magnet and just a few pieces of metal were collected, enough to make a stick pin. Will have to clean it up to see better, just haven't gotten around to that point yet.
Where is the best and least expensive place to purchase replacement parts for the transaxle and what is the best grease to put back in. How much grease and is there a better grease to use for winter use, pushing snow?

I think that might be a Peerless 820 series.
Download the manual at the link and see if it looks right

Here is a link that might be useful: The Link

I had a similar issue with Sears Craftsman 5000 6 speed 25 horse garden tractor.

Put it in gear an it wouldn't move. Checked drive belts, inspected gear linkage, drive belt tension. Mower deck operated fine. Checked all electricals suspecting possible electric clutch issue and finally settled that it was stretched drive belt, the woodruff key on the transaxle or a failed transaxle. I started to research cost of transaxles and discovered that searching by the replacement part number, I could find a new transaxle at half the cost of Sears parts. Then one more idea came to mind, the range selector lever might be disengaged. Went back to the machine and sure enough the range selector was half way between high and low range, disengaging the transmission. Ironically this lever is rather obscure in the user manual documentation and the ground drive parts schematic. My headache is gone and I saved over $700.00 in parts.