UTI was my first thought, too. It wouldn't hurt to have that checked since there can sometimes be a UTI without obvious signs such as a change in color or odor of the urine.
If he was always urinating a lot and now is urinating even more, I would suggest that you ask the vet about checking him for kidney changes. For example, my dog developed renal insufficiency when he was middle aged and began to drink so much that I had to buy him a larger water bowl. I was changing his pads several times a day and we were using Serenity Ultimate pads which were the high absorbency ones. Some people here have used baby diapers (instead of pads) laid flat inside the malewrap for added absorbency. Dogs can live a long time with renal insufficiency if you adjust the diet, but you need to find out whether he is concentrating his urine.
Another condition that causes increased drinking is diabetes. You might just want to get him in for a general check-up, or if he is a senior get a senior wellness check-up.
You were asking if increased incontinence can gradually happen over time. I do not know the nature of the spinal injury for your particular dog. Beagles do get IVDD which is a disk problem, and I am guessing that is what you were referring to, and it sounds like (?) your dog is walking? Have you seen any change in his mobility? In general, any dogs can develop incontinence as they age, due to age related changes, so what you are seeing now could be something different than the original disk problem.
The best way to handle fecal incontinence in dogs is a combination of diet and expressing the bowel. Diet can create firm, low-odor stools that do not harm the carpet and will not make the house smell bad if he doodles while you are not home. Murphy was a beagle and Murphy's Dad used to express his bowel by putting an ice cube up against his sphincter. I've done that with my dog, too. Here is an article with more detail on handling bowel incontinence in injured or aging pets. viewtopic.php?f=5&t=18586