At least on the spin bikes at my gym, the pedal tension isn't adjustable like it would be on a bike set of pedals. Or if it is adjustable, it's somewhere not easily visible by inspection. So there's really not that much recourse other than to be more careful.

Since the pedals on a Spin bike are just spd-compatible, not actual shimano brand pedals, they were probably designed to take a wide range of SPD-like cleats, such as wellgo, etc. It may be that a different type of cleat would work better, though I'm not sure. I actually do spinning with cannondale MTB shoes that I bought years ago, and have had the same cleats on since then. The cleats are from 2002 and double release. I use the same shoes/cleat for commuting and club rides, so they get a lot of use indoor/outdoor, and hold in a spin bike fine. Try swapping to a different cleat, maybe even to an off-brand one, and seeing if that works better for the spinning.

You may wear your cleats faster, and if you have road-style shoes with plastic or carbon soles, you may scuff them; but otherwise there shouldn't be any real damage. Try different bikes in the classroom; where I go they never move the bikes so I've already gotten used to which bikes are good and which ones are bad (mostly have clicky bottom brackets, but one or two have a missing pedal cleat on one side).