I'd say it depends on a few things. First, are you in an area that has other universities/do many students from Shenandoah live off campus/are there a lot of hospitals near the area ? If so, then you may need to start looking sooner because of that competition. Also, if you find an actual apartment complex instead of just renting a house or something then often times if they have someone leave before their lease is technically up they'll allow you to move in when you want to and not charge you for signing the lease in advance. Basically, the person who leaves the apartment before their lease is technically up might have to pay rent until you move in even if they're not living there (hopefully that makes sense?). I just recently signed a lease with my roommate for an apartment in Syracuse because our program starts June 1st and that's what we were able to do, we obviously had to pay the security deposit up front, but we don't have to pay for June until the week before we move in.
I'd maybe start by calling around to different places and find out how early they typically sign leases for August.