There are a couple of things you can do on the dragging. First, you want to be sure he isn't licking or chewing himself. This is a temporary phase some dogs go through when their nerves are recovering from paralysis. The theory is that they are feeling a pins-and-needles sensation, similar to what we feel when we sleep on our arm funny and it goes to sleep, then begins tingling when it wakes up. Or something along that line. Some dogs will lick and chew the area until they can actually do damage, they don't seem to realize they should stop. Some dogs may lick or chew their feet or other areas, but the pen*s is definitely one place that they may lick/chew. If you think that is what is going on, there is a medication that works extremely well, called gabapentin, and I would get him on it right away. The cone collar may be a temporary fix but dogs can get out of it.
Is it possible he lost weight and his wrap doesn't fit as well?
There are wraps with suspenders, but if you think he has just become more active, then I would go with a drag bag. You can find different styles made by different companies. Here is an example:
https://www.handicappedpets.com/walkin-drag-bag
On the question of recovery, there isn't really any way to know for sure. There have been dogs here who had a return of deep pain sensation, bladder control, and walking many months after the injury. And of course there are dogs who remain incontinent and use a wheelchair, but they still have happy lives.
Cindi's dachshund had deep pain return after 9 months.
viewtopic.php?f=4&t=5673
Marilyn's Wally had no deep pain and didn't walk for 18 months but learned to walk again, and he began to recover bladder control even after that.
viewtopic.php?f=4&t=9549&p=48548&hilit=wally#p48548
Joanne's Tiny began walking after a year and doing stairs after 2 years.
viewtopic.php?f=4&t=18657#p96289
Syd began to take steps and urinate on his own after 2 years.
viewtopic.php?f=4&t=18791#p100757
Rehab is helpful if you can afford it. You can also do rehab at home, though I don't know how heavy of a Bassett he is, so it depends on what you are able to do. Swimming is great. If you have a pool or can take him to the lake, that would be great fun, great exercise, and helpful with recovery. Swimming also means less effort for you (besides getting him into and out of the pool, or in and out of the car for a trip to the lake.)
I wish I could give you a more definite answer, but the truth is, you just don't know.