The dash does not need to come out. The heater box behind the dash does need to come out to access the core. If you have factory in dash A/C, the process is more involved as the A/C needs to be disconnected under the hood to get the heater box out. With no A/C, removing the glove box should give you the access you need to pull the heater box.
As an aside, you can disconnect the heater core from the heater hoses under the hood, and re-route the heater hose back to the engine, bypassing the leaky core, thus giving you the opportunity to drive the car (without further leaking problems) until you find the time (and $$) to have the repair made.