I wear polarized Bolle sunglasses every single day. Literally every single day, dawn until dusk (I have a raccoon tan unfortunately!). I've never experienced any depth perception problems at all. Before I bought my sunglasses, I looked at some research that suggested polarization is bad for golf, but other sources disagree. I'm really not sure who to believe, but in my own personal experience, polarization does not affect depth perception.
That's because "polarization" is not synonymous with "UV protection." They're two mutually exclusive terms, and anyone looking to buy a nice pair of sunglasses should know the difference. UV protection (A,B, and C) is essential in a pair of sunglasses IMO. You've only got two eyes, you should probably protect them as best you can. Plus, I see much better in sunglasses. I think sunglasses are a great idea, and you definitely get what you pay for.