If the crystal is running hot, it means your drive level is, well, excessive. You could also have some DC present that shouldn't be there. You can order crystals in different packages, HC-49 and HC-6 are the most common.
Taking a look at the circuit, it appears this transmitter is a power oscillator coupled to an antenna. The crystal is either not suited for this purpose (microprocessor crystals usually aren't happy in these kind of circuits) or you may have the coil the wrong way around, driving the crystal too hard.
+-RH