Hello, and thank you for your question about your Pug. I hope I can be of some help.
The fact your dog is wobbly and weak with walking, and has "uncontrollable bowel movements" (what vets refer to as "fecal incontinence") does point strongly to a neurological problem, specifically a spinal problem.
The hard part with that is that there is often little that can be done to fully fix the problem, but there are some things that may be helpful.
There are a number of spinal problems that could cause what you are seeing. Probably one of the most common is intervertebral disc disease (IVDD):
You will see fecal incontinence mentioned at the end of the article. Sometimes back surgery can help these dogs, but otherwise, anti-inflammatories can help bring down the swelling/pressure on the spinal cord and could help with some of the symptoms you have described. This may be what your dog has.
Dogs can also have severe arthritis of the vertebrae (spondylosis) and specifically they can have arthritis and narrowing of the spinal canal (which puts pressure on the spinal cord and nerves) where the spine connects to the pelvis. We call this lumbosacral stenosis. This also can reduce nerve sensation to the rear end and lead to rear leg weakness and fecal incontinence.
If you want more specific answers on what is occurring with your dog, spinal x-rays would be the first step. You could even be referred to a specialist for more advanced imaging tests like an MRI, if desired. If back surgery is an option for your dog, that could be explored, or at least there may be helpful medications.
Here is some information on managing paralyzed dogs. I know your dog is not paralyzed, but some of the tips mentioned here may be helpful now or in the future. His disease could possibly progress, but with a better diagnosis and treatment, hopefully he would not worsen and may even improve:
I hope this is helpful information. Thank you so much for your question, and I hope your pug will improve. They are a wonderful pet!