I am in dire need of some help.

I recently divorced (still going through it) my wife of (almost) 2 years. I moved over to the USA to live with her.
I am still in the USA and my permanent resident card is ticking down, I only have until March to stay here.

My worry is, I have fallen in love with the most increadible women I could ever imagine, and now I am worried what plans should be made from here.

My first question is: if I overstay my visa for any period of time, will I be refused entry back into the USA?? Will I be Deported?

My second question is: Does anyone know of any way that I can extend my stay legally? My girlfriend has two dogs and we plan on moving out of the USA when we can, but the time it takes to vaccinate them overruns my stay.

Thank you so much for any help.
P.S.
If i'm in the wrong forum, let me know :-)

You go first and have your GF follow.

You should be posting in the forum called "Effects of Major Changes in Family Situation on Immigration" or something like that.

You can self-petition the I-751, basically file it alone during the 90 days before your green card expires and try to prove to USCIS that even though your marriage didn't last, you did come here in good faith.

Yes, AFAIK if you stay here while out of status you are risking a ban. How long the ban lasts depends on how long you're out of status.

Go read over that forum, they know far more about it than I do.

But if you don't want to stay here permanently, it would make far more sense to do what LuckyStrike suggested.