Can a case act as a shorted turn for a coil?
Should I add an air gap, by adding a groove down the side of the cylinder [and re-filling with epoxy]?


No need to add air gap (slit) down the side. There is not a gap that you "can't see" in the IF shields (IF cans).

In the toroid transformer, the magnetic field lines ALL go through the "shorted turn", regardless of the shorted turn diameter (with some minor leakage), because the core has no beginning nor ending (no gap). Thus a true shorted turn with negative results.

For IF coils (transformers), envision this : the Core on which the transformer is wound is a short cylinder (very long air gap). The magnetic field lines leave one end of the core and return to the other end of the core within the shield. Thus, very few magnetic field lines go through what looks like a shorted turn (shield).

If you made a shield can diameter to be very small (close to the transformer), then more and more of the magnetic field lines WOULD go through the closed loop shield can making it more and more like a shorted turn. So, don't make your shield (cigar tube) too tightly coupled to your transformer.

Note of worry  : It sounds like you are going to use a "loop stick" type of coil (transformer) from an AM radio.  The length of this type of core (a long cylinder, not a short cylinder) will require a larger diameter shield box, so that the magnetic lines have room to exit one end of the core and re-enter the opposite end of the core (within the shield). I do not have a rule of thumb to help with deciding upon the proper clearance of the shield as relates to the length of the cylinder.

In reality, I don't see the need for a metallic housing as regards performance. You could use a non-conducting housing.

EDIT :  Add illustration