You can apply for a green card while on TN. While your application is pending, there is no problem. If you apply for advance parole, there should be no problem leaving and re-entering the country. If the application is approved, well, you are on your way to a green card. However, if your application is declined for any reason, you have a problem. You've shown immigrant intent by applying for the green card and that is incompatible with your TN status. For this reason (the possibility of being declined), it is typical to convert to H1-B first. Because H1-B allows "dual-intent," it remains valid even if you're green card application fails.
TLDR: you can apply for a green card while on a TN, but it is risky if you're application gets declined.
Your current employer should hopefully be willing to sponsor you for a H1-B and ultimately a green card. If not, you might wonder why not.
OTOH, I don't think they actively seek you out to revoke your TN status, but it would make border crossings pretty much a non-starter.
All this is remembered from when I was on a TN status from Canada ~5 years ago.