Green card through marriage with an expiring soon J-1 visa

[Rom177]

Hello everybody,
so my situation is a little bit tricky, let me explain. I'm a French citizen and came to the USA 4 years ago through a J-1 Visa (Scholar (I have a PhD)) program. I'm a Postdoctoral researcher at a public institute (University of California San Francisco) and my visa is about to expire at the end of the year (December 2021). At this time, I would have reached the 5 years limit of the J1 visa. 3 years ago, I met my partner (who is an US citizen) and we are planning to get married. Since my visa will expire, we are planning to fill a petition for me to get a green card. However, since the average timeline to get the green card is 10/13 months when you marry an US citizen and you are already in the US, I would get my card after my J1 visa expire (FYI, I'm not required to go back for 2 years in my country). One of my first question is, is there a way to get a temporary form between the end of my current visa and the final obtention of my green card to allow me to stay and work in the US? The other solution could be to get another type of visa (like H1-B, since i'm working in an academic institution, i'm not subjected to any cap), but is it possible to apply for a visa when you are already applying for a green card ? It's all very confusing so if someone could help me, I will be really grateful! If you need more details, don't hesitate asking!
Thank you very much!

[newacct]

First, don't confuse visa and status. Visa is just for entry. Assuming you were admitted with D/S on your I-94, you remain in status for as long as you maintain a DS-2019 and comply with the requirements of your program; you also remain in status for 30 days after completing your program.

You can stay in the US for as long as your Adjustment of Status (I-485) is pending, regardless of whether you have status or not. For all categories of I-485, going out of status while I-485 doesn't affect the I-485. For your category, even going out of status before I-485 is filed doesn't affect the I-485.

Together with your I-485 you can file I-765 for EAD and I-131 for Advance Parole, both for free, to work and/or travel when your I-485 is pending. Once you get an EAD you can work. Note that EAD/AP may take more than 6 months these days.