IANAL. I have been on TN status a couple of times as well as H1-B. Pertinent facts: TN is strictly temporary; you are not allowed to have immigrant intent. H1-B, while temporary, allows you to have immigrant intent. So, the situation as I understand it is:

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.

IANAL, but am a Canadian. My belief is that TN status is a "non-immigrant" status. It is, by definition, temporary. Applying for a GC declares immigrant intent. H1B, by contrast is 'dual-intent'.

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.

how do border crossings go for TNs. I'm about to get one, and am planning on visiting back reasonably frequently. Is that a problem at all?

My friends recommend signing up for the Nexus program to facilitate speedy crossings. As I understand it you are effectively re-applying for your TN at every re-entry and could be rejected. That said, my friends all go back to Canada many times a year without issues.

Not a problem in practice. I crossed the border dozens of times with a TN (I actually lived in Canada but worked in the US) and know of many others. Generally as long as you aren't in violation of the intent of TN you should have nothing to worry about.

I had a similar situation (I am Canadian and was on TN). I successfully applied for a status change via marriage on account of my wife being a US citizen. My attorney said that to get around the temporary intent of the TN status, I had to wait at least 2 months after entering the country before applying for the change of status, in addition to her legal footwork. This meant being able to stay in the country for at least 8 consecutive months (2 months wait, up to 6 months for travel authorization documents), which may be complicated for some. It is possible though.

I know personally that the TN is a non-immigrant intent visa. Typically you have to switch to an H-1B ("dual intent") first.

I would need to find a workplace that I love first

I'm in a similar situation, except Canadian. Did it take you 6 months to get your wife's TD status? I applied for my wife back in March and we are still waiting for her TD (she is not Canadian). :(

Woah! My wife and me got it together. She came in 6 months late because moving was an ordeal (we have two big dogs).

Yeah we had to apply for permission for my wife to enter the US to ask for her TD status. It was supposed to take 7 days but we are still waiting.

Did you applied remotely? That usually takes a bit longer.

I went to the US first to get my TN, before returning to London, UK where we were living. We then applied at the US Embassy for my wife's "pre-TD" if you will. The consular officer there told her she was approved and would get her passport back in 7 days. Then we discovered her application is in Administrative Processing, and we've been waiting ever since.