Shimano Shadow vs Non-Shadow Rear Derailleur for Commuter Bike
Shadow describes the design that brings the derailleur body inboard making it less prone to knocks and damage, it does not help with chain slap. For that you need a clutch or in Shimano phraseology "Plus"  (SRam "type 2")   

You can get Shadow derailleurs without the clutch (e.g. M592), so if you want the clutch, look for a "plus" - I believe the lowest in the range with a clutch is Deore.  

Without getting into buying advice (Off topic here), the difference between components one level apart is small enough most people will not notice. My choice would be based on internet shopping for the best I could get within my budget as often you can pick up great discounts. 

Shadow is not a big deal unless you regularly wipe out you derailleur in rock gardens, but I would pay a few dollars more for a shadow Plus if you are noticing an annoying amount of chain slap, but from the sound of it a Deore will be more than you want to spend.   

If you are on a tight budget, also have a look at Microshift derailleur, don't think you have to stick with the $himano tax.