(This is not legal advice. Consult a securities attorney.)
> you receive a physical stock certificate that you should be able to do as you please with...
No. The shares you’ll receive after exercising almost always have transfer restrictions. To sell or otherwise transfer them, you’ll almost always need the company’s permission.
Some companies are okay with it, others aren’t, others decide on a case-by-case basis or have specific requirements (like not increasing the size of the cap table). Read the options agreement and then ask the company, all before exercising (and potentially incurring a taxable gain). If possible, do all of this well before leaving.
(What you’re describing is closer to publicly-traded securities, which you mostly can “do as you please with” in terms of dispensation.)
