To swim, simply look in the direction you wish to move, and move forward.
It might also be benefitial to pick up issue 7 of the Wasteland Survival Guide, found in Old Gullet Sinkhole, as this will reward you with a perk that increases swimming speed by 25%
I believe the issue you are having lies elsewhere, in regards to this quest.
The first thing you want to be doing is bolstering your radiation resistance. I watched a level 20 friend complete this, and the first thing he did was use some Rad-X. It significantly decreased the amount of radiation he absorbed. You could also go for more permanent protection, such as lead armor, or a reinforced Vault 111 jumpsuit. It might be handy to carry a Rad Away on you, just in case, and have it mapped to the d-pad.
You will notice an elevator of sorts, further down past Sully. If you press the button, the elevator will rise up, although it takes a little. Inside this elevator is a lock box with a Novice lock, holding a Rad X.
The second thing you want to be doing is paying attention to those bubbles. By identifying the bubbles forming at the top, you can identify where the leaks are, before having to dive in. Furthermore, the bubbles emit directly from the leaks, allowing you to dive down and fix them quite quickly.
The valves are yellow-coloured, and stand out from the pipes. If you follow the pipe directly down, they are impossible to miss. You simply have to use them, there is no hold down or turn animation to wait through. The second I hit the valve, I am moving on to the next, or surfacing for air.