If you rent an apartment with a lease but have to move, do you have to pay for the full lease term?

You would remain liable for the rent for the balance of the term of the lease or until the place is re-rented.  Once the place is re-rented, your obligation to pay rent ends.  The landlord cannot allow the rental to remain vacant for the balance of the term of the lease without making reasonable efforts to find another tenant.  If the landlord does not make reasonable efforts to find another tenant, the landlord has failed to mitigate (minimize) damages (the amount the landlord claims you owe) and lthe landlord's damages will be reduced accordingly.  Reasonable efforts on the part of the landlord to find another tenant will be determined by what other landlords in the area are doing to attract tenants; for example, posting a sign on the premises advertising the vacancy, advertising the rental in the newspaper, online or in a local rental guide, etc.

Once the apartment is re-rented, your obligation to pay rent ends; however, if the new tenant is paying less rent than you were paying, you may be liable for the difference in rent for the balance of the term of the lease.  The landlord must have a valid reason for charging the new tenant less rent, such as market conditions.   If the landlord does not have a valid reason for charging the new tenant less rent than you were paying, the landlord has failed to mitigate damages, and the landlord's damages will be reduced accordingly which means you may not be liable for the difference in rent for the balance of the term of the lease.