How can I reliably catch fish without drowning?

Just blow them up.

No, seriously. The ol' 'Dwarven Fishing Pole' is alive and well in Zelda: Breath of the Wild. If you throw a bomb into the water where the fishes you want are, and detonate it, they'll rise - belly up - to the surface. Then just dive in and collect your seafood at a leisurely pace, without using the dash button and swim to shore. You can even jump in multiple times as needed without any fear of losing 'The One That Got Away'.

Those fish ain't going anywhere.

As a less fun alternative to blowing the fish up, you could simply create an ice block in the body of water you are fishing in for a resting spot for when you get low on stamina. Then dive off, search for fish, and clamber back on top of it for a breather.

I haven't tried this, but I saw a tip on a loading screen. You can apparently drop food in the water to attract fish. Which fish are attracted to what food is a matter of experimentation, however.