No, you don't remove the adrenals. That's because the adrenal medulla would be removed as well, the source of catecholamines, and this would be a Bad Thing. However, if the cause of the Cushing's is an adrenal tumour, this is probably unilateral, so removing the diseased adrenal only is a practical course of action - like kidneys, you can get by on one.
Most Cushing's cases however are pituitary in origin. In man, one of the preferred treatments is removal of the pituitary lesion. The original Cushing was a brain surgeon! This is not done in dogs.
There are two drugs (well three but we can forget ketoconazole because it's not a good idea) which will affect the adrenal cortex.
Mitotane is the older drug. It's never had any sort of product licence because it's not very safe (its effect is irreversible), but it can be used two ways. You can use a moderate dose to get adrenocortical activity down to acceptable levels, and then usually stay on maintenance treatment and this was done quite successfully for many years. Or else you can use a high dose to obliterate the adrenal cortex entirely and then maintain your patient on Addison's maintenance therapy. This latter was used more in Europe, and I've seen some good survival rates, but there were a lot of problems getting the right dose in the first place.
Trilostane is the newer drug. It is reversible in action (if you stop the drug the adrenals recover), and so safer, and so it is licensed. It can only be used for long term maintenance, as you can't completely nix the adrenals with it. Because of these factors it got a product licence, and now vets have to use it as first line treatment in all cases. If you feel a patient needs mitotane you have to go through a lot of regulatory hoops to get authorisation to import it as an unlicensed medicine. However, published survival rates are just about the same as moderate long-term mitotane maintenance in fact.
Last I heard of it, trilostane didn't have a produce licence in the USA, and there was a thriving black market.
Rolfe.