The differential diagnosis for drinking more (and urinating more) is fairly extensive. It includes diabetes mellitus (origin lack of insulin), diabetes insipidus (pituitary disease) kidney disease, adrenal disease, treatment with glucocorticosteroids (prednisone), increased salt intake, diuretics (Lasix), liver failure, low protein diet, hypertyroidism (cat disease) or administration of thyroid hormone supplements, high blood calcium levels (some types of cancer), other electrolyte imbalances, severe systemic bacterial infections, high ambient temperature/humidity, brain lesions in the thirst center of the hypothalamus and psychogenic (habit).
I suspect that the basic blood work would rule out some of these (diabetes mellitus, kidney and liver disease, high blood calcium and other electrolytes). You know if you are giving corticosteroids or diuretics or if it has been very hot where you are. Because the tests included in"basic blood work" are variable depending on who is doing it, these diseases may still be in the differential.
It would not rule out diabetes insipidus, adrenal disease, increased salt intake, low protein diet or malabsorption, thyroid hormone supplementation, bacterial infections (cystitis), some electrolyte imbalances, bacterial infections (cystitis), thirst center brain lesions and habit.
Increased salt intake can occur from being fed salted/spicy people food, licking livestock salt blocks, ...
A urinalysis would help differentiate some of these. Very low urine specific gravity (<1.009) would point toward diabetes insipidus and a look at the urinary sediment would detect bacteria and abnormal cells.
The thyroid supplement started after the increased drinking, but could have increased it further accounting for the bed wetting and not being able to hold urine for as long as before.
It the blood urea nitrogen (BUN) was low, renal medullary washout should be considered.
Adrenal problems can be diagnosed by looking at some of the blood values and doing follow up studies to see if the adrenal responds to stimulation.
There are flow charts in veterinary medical books for diagnosing causes of increased water intake/urine output that are very useful particularly once the common causes are eliminated.
Let me know if you have follow up questions.