Unfortunately, many dogs with Addison's disease are treated with too much glucocorticoid. Remember that our goal with glucocorticoid supplementation is to provide the same amount of steroid that the dogs would normally produce if their adrenals had not failed.
For dogs, the daily glucocorticoid maintenance dose for prednisone is only 0.1-0.2 mg/kg/day (3), so that calculates out to only 0.5-1.0 mg per day for Scooter, quite a bit lower that what you are currently giving (2.5 mg per day). That would certainly be enough to cause an increased thirst by itself, but would also contribute to glucocorticoid-induced insulin resistance, making the diabetes uncontrollable despite the higher insulin doses.
Therefore, we should try to lower the prednisone dosage first down to 1.0 mg once daily (or divided). If he is doing well clinically (i.e., normal appetite and no vomiting), then the dose can be lowered even further, down to 0.5 mg per day. Prednisone or prednisolone are available in 1-mg tablets, as well as an oral solution, making it possible to administer these smaller dosages (6,7).