I just had a very similar dilemma. We bought a king bed and all new sheets. Splashed out at the sales and got 1200 thread count. Then I looked at the tag and they said no tumble drying

I'm someone who puts pretty much everything in the dryer all the time . So there was no way these weren't going in too. I do put them on low heat and so far they have been fine. So I tend to think if you do them on low you should be ok.

Also, I've found with things that have shrunk in the past (not sheets though), that if I wash them again and air dry them they go back to the original size. So I guess I'm thinking that if I do end up shrinking my sheets I may be able to resurrect them (might be wishful thinking though). HTH