No, I always dry my clubs during play, and especially after play in the rain, and have never seen any rust on my shafts. I guess if you just bung wet clubs in a locker, it mat be an issue. I always take mine home, and if wet, line them all up against a radiator.
Chrome is a decorative finish, not a rust protection. It is porous.
Murph, I clean and use Yes! wipes on my clubs every week and if they get wet I do the same as you. My FP's shafts have no pitting but the Srixon wedges have some pitting on them, the reason I thought this is because the FP's are Nippon and the Srixon are TrueTemper and both have been cared for the same as only a loving parent would.
Due to a lack of garage and garden shed space, my clubs spend most of their time in the boot of the car. Left them in there for ages before Christmas, took them out and they had got a bit damp. The Nippon shafts in my irons were ok with a wipe down but I noticed that the shaft in my YES putter had a few small rust marks appearing on it. Managed to get rid of them with a little elbow grease but will be keeping an eye on it.
The slimey grip took a little more to sort out though and the smell from an old glove in the ball pocket had to be sniffed to be believed...