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Yes, there are certainly a range of drugs for treating incontinence in dogs (including DDAVP), however the incontinence is likely due to the steroid solely - hence changing the drug from a steroid to a non-steroidal would solve the problem here. Giving an incontinence med alongside the steroid is probably unlikely to help as the issue is increased urination as a result of increased drinking.
As such, changing the steroid to the nonsteroidal drug will likely do the trick here. Once your vet has a definitive diagnosis about the tongue lesions, then we will know whether the steroid is needed for this or not. If Charlie must stay on a steroid after that, then adding in tramadol or gabapentin alongside the steroid may allow the dose of steroid to be reduced such that there is no urinary issue, but the arthritis is still controlled.
Definitely follow up on the above with your vet when you get a chance.