Are you competent and qualified to accurately test this motor? What are you using to test it and how? The reason I ask is this:
A first year apprentice should already know that you will read continuity from hot to neutral through the motor windings themselves, and neutral and ground are SUPPOSED to be connected at the service, so of COURSE they would read continuity. Basically, what you have posted as evidence of a possible issue means nothing.
So giving you the benefit of the doubt as an apprentice, the main lesson here is, if you don't know what you are doing yet, don't take it on yet.
Other lessons for your growth as an electrician:
1) Motor health is checked with a megohmeter, often called a "megger". It charges up a DC voltage with a capacitor from a battery, or a hand cranked generator, and you look at the resistance between the windings and ground as that DC voltage leaks through the winding insulation to the case, looking for it being less than a certain value (different for different motors, but the megger will come with a guide book) to know it is bad.
2) Putting your hand on a motor is not a valid determination of motor health either. Motor insulation is rated for MUCH higher temperatures than the human hand can even BEGIN to stand. For example, Class H insulation, common on modern energy efficient motors now, is OK at temperatures of up to 357 degrees F. Your hand will jerk back at somewhere around 120 F. So again, that means nothing.
So aside from these two invalid results, was there any other reason for you to suspect a problem with that motor?