IANAHP (healthcare professional), this is not medical advice, I know more about animals than I do people (although basic physiology is similar in mammals), etc.
What goes in, must come out, one way or another. Assuming your insensible losses (such as from your skin, mucous membranes, and lungs) and sweat output have not dramatically changed, drinking more is going to make you pee more.
The technical term for drinking and peeing a lot is PU/PD, which stands for polyuria (peeing a lot)/polydipsia (drinking a lot). They generally go together, due to aforementioned laws of conservation of matter.
The question is, are you peeing more because you are drinking more, or are you drinking more in order to keep up with peeing more?
Drinking more than you need to (polydipsia first) is usually psychological, i.e., you are drinking more just for the heck of it, because you like what you are drinking, because it’s a nervous habit, etc. I drink water constantly at restaurants, waaaaay more than I need. It’s fine until I get in the car to drive home and have to go 5 minutes after we leave the restaurant. Some mentally ill people, if not prevented, will drink themselves to death from hyponatremia. I have seen horses who drink excessively in the summer (psychogenic polydipsia) – their water needs are somewhat increased due to increased losses, but a few horses just won’t stop, going through multiple buckets a day of water and flooding their stalls with urine. The urinalyses, kidney function tests, and metabolic panels all came back normal on these horses and the behavior waned in the fall.
Lots of different things could cause polyuria which dehydrates you and causes polydipsia to compensate. Kidney disease is a big one in cats. Liver disease. Diabetes insipidus is a lack of ADH (or lack of response to ADH) which keeps your kidneys from concentrating urine. Metabolic disorders, such as Cushing’s disease, can cause PU/PD. As you mentioned, diabetes mellitus can cause PU/PD - the glucose in the urine pulls water into the urine by osmosis. Electrolyte imbalances and some drugs can cause PU/PD.
Waiting for QtM, Pullet, or someone else to come along and give a real answer, as opposed to my half-cocked ramblings.