Can my unit restrict me from taking a certain amount of leave before a PCS?

It's a PCS. Your commander is the approval authority for leave. You're not automatically authorized any amount. You're only authorized travel time.
Out in your leave form plus request PTDY. Assuming you have 30 days of leave, you can send off in April.

You can request leave. The commander can deny it.
If you are wanting leave prior to the PCS solely to avoid NTC, or to take a vacation prior to reclass, the needs of the Army (unit's NTC mission) takes priority.
However, it may be worthwhile to contact the gaining installation to check on the availability/average wait time for housing, and then the school so you'll know what the process and timing normally experienced (save the name and phone number of any contacts). This may provide information to support your request, of your intent is to arrive early enough to search for, and secure, housing.

Your command has the authority. There are limited authroizations based on unit policies. Often times a waiver can be had. Just request and see. You have no recourse if disapproved. Thank you for your service.