Ray,
Do you have a factory service manual (FSM)? I looked in my '71 GT FSM which has an extensive section on the automatic transmission. There is a diagnosis table and the possible causes for the "No park" condition are:
a) selector lever linkage improperly adjusted
b) parking lock actuator spring
c) parking pawl
d) governor hub.
Selecting park moves a rod inside the transmission which pushes a hook like piece (the parking pawl) into the notched outside diameter of the governor hub, thus locking the output shaft from moving. The rod has a cone shaped piece on it that the spring (b) pushes back. The cone moves the pawl into the governor hub. This cone is spring loaded since the hook doesn't always line up with a space in the hub when you stop the car so when park is selected the car may roll forward or back (only fractions of an inch are required) until the pawl engages. From the diagrams it seems b), c) and d) above can only be checked by transmission disassembly. The pawl is also spring loaded to keep it retracted from the hub, this in your case seems OK since you have no park lock.
Since you said your shifter is sloppy maybe a) is the cause? There are adjustment instructions in the FSM.
If it's an internal thing, I would be worried about parts/pieces inside the transmission. Therefore towing rather than driving would be the safe bet.
Sorry for the wordy response, but hopefully it helps. Here are some scans from the FSM to help with the words:
Auto Trans Cut-Away.doc
Auto Trans Linkage.doc
Harold.