Not a problem. I work with 20-40 new students every semester so I see
lots of underwater swims...you're not the first to be dense.
Personally I prefer being dense, I wear less weight that way.
I'll assume this is for your GUE courses?
Without the pushoff, you lose the momentum, but you gain a bit of realism. There are no sides in the ocean...
To do it without the pushoff, personally I'd drop down, do an arm pull to get a bit of momentum going, and do exactly what I said above.
As to exhalation drills--I'd do them first, even in the cenote. Helps get you nice and relaxed in the water...I'll be honest, I didn't believe in them when I took my basic scuba course. Then as I started volunteering, it was expected that all the volunteers would do the whenever we go in the water as a good example for new students. I did them out at the quarry in cool water while helping with new student checkouts, and only then did I realize how well they work. At first it's hard to breathe (especially if the water is chilly), but after a few forced breaths off your regs with your face in cool water, you feel completely relaxed and breathing is now totally natural...it probably means a lower SAC rate as well, although I haven't conclusively proved that.
This would also probably be a good thing to do before your Tech 1/2/3 courses since they apparently do lots of maskless ascents/mask-off drills...
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