When DS was that young, I didn't bother with the shower there, I always saw Mums traumatizing their LO's with baby screaming, Mum frustrated, so that I vowed I would never do that! We would bath together once home in the comfort/familiarity of our house straight after. (but once he was walking at 10.5 months, we started showering/rinsing off there btw)
I would dry him off the best I could out in the pool area as the changing rooms are even chillier! Then I would dress him in real cozy, warm clothes. I found it handy at our club they had a playpen so I could lay him down once dressed on his blanket playing with his little soft toy things whilst I got dressed, dried my hair sometimes, if not towel dried etc...do they have that? You need to be hands free. You could bring your car seat in too for after when they are ready to chill and be carried away, always a good nap in car after too!
I brought hats for us both in autumn/winter/spring months as hair is damp for going out to car.
Have fun.