Is Wolverine Logan immortal even if he drowned?

Yes. Given the many skills Logan has learned over his long life, he would have learned and used swimming while he was a member of covert operations. For most protocols this would be a requirement.

It's not a question of training or skill

Of course, Wolverine can swim. His body weight, although an impressive 325 pounds, should not be able to compete with superhuman strength and endurance. He is capable of easily lifting 800-1200 pounds and running at speeds of up to 35 mph for hours. His body produces limited fatigue toxins and these are processed instantly so that he can swim almost indefinitely. The operative word is almost. Trapped at sea, even he would drown from exhaustion.

His body, fortified even with his extra 105 pounds of adamantium, would be a perfect improvement as he is at the core level of his being. He would work a little harder, swim a little harder and his buoyancy would be compromised, but he would use his raw muscle strength to overcome it. We have human SEAL team members trained to swim with 100 pounds of unbalanced and awkward equipment. His weight is fully internalized and already balanced.

Wolverine's problem with water is psychological.

Warrior, berserker, engine of destruction

Logan is the living embodiment of combat. His body is filled with a genetic disposition to fight. Bones and tissue regenerate after almost every injury, as quickly as it can be damaged within reason. A body with bone claws, superior physical senses and a fighting consciousness that has been sharpened through decades of training. Beserk rages and makes his body even more powerful and deadly under duress. Even if he is overwhelmed by enemies, he can keep fighting!

Swimming, and especially drowning, are directly opposed to its natural tendencies. Drowning is the opposite of struggle. You are alone in the dark and have no one to direct your anger against. Oceans are large and can hardly be crossed on their own.

Now imagine that you are a man whose body is designed to fight and win against any threat, surrounded by the only enemy he can never defeat, no matter how immortal he might be, this would be an enemy, that he would face mentally unhappy, no matter how little or much he had to swim.

Innate fear exacerbated by its natural ability to survive

Wolverine hates water because it can kill him over and over until he is rescued or until the trauma of drowning, one of the most traumatic ways to die, drives him insane. Drowning goes against his nature, he is a being designed to fight, and he is used to fighting effectively against any enemy. Fighting while drowning only increases the rate of drowning. Based on the way he speaks when he talks about swimming, I can assume that he has experienced this trauma at least once.

Yes, Wolverine can swim. He does this with concern because it is one of the few ways he can get killed over and over and is totally dependent on someone else to save him. As the perfect warrior, his psyche has trouble processing this, and he is, and it makes sense, that he should be water averse.