Shavings (or whatever) do not temporarily rob N from the soil if used as MULCH, only if they are actually mixed in. So just plain ol' shavings are really pretty safe as mulch. Also, even if you do stir them in, the fact that these are USED shavings means they will be bringing at LEAST as much N as they're likely to need WITH THEM, so that is not likely to be a problem even if you did mix 'em in rather than use as mulch.
The real considerations IMO are slightly different.
1) if your litter is REALLY pooey, it may have an excess of N that will easily leach down into the soil when you get rained on, and has the potential to damage plant growth by an excess of N. You *can* get away with putting fresh chicken poo or pooey bedding around *some* plants in *limited* quantities at *some* time of year but you need enough gardening experience to be able to make reasonably-educated guesses about where to draw the line, AND I would never recommend doing it around perennials or (worse) trees/shrubs that you would be highly distressed to lose.
2) You can get E coli (and other food poisoning type bacteria as well) illness from eating veggies that have got too-fresh poo against them, either from lying onto the ground or from rainwater splashing or whatever. Certified organic operations that use manure are legally required to age it for I believe it's 60-90 days before use, for this reason. The risk is obviously much greater for things like lettuce or spinach, whose entire growth cycle occurs within that time span and are right down there on the ground getting dirt on 'em all the time, as opposed to things like corn where even if you planted into recently-manured ground the ears won't be mature for several months and anyhow they are firmly held quite a ways up above the ground.
Obviously it's a personal decision how much you gonna worry about food poisoning from fresh manure on the garden, but it is a consideration to at least be aware of so you can make an informed decision rather than being surprised if something goes wrong
Good luck, have fun,
Pat