Advice on changing status from J1 to marriage green card

I’m just looking for some advice and insight into my case. I am currently visiting on a J1 work visa, I arrived at the very end of July this year, employment started on the 2nd of June and my last day will be April 15th. My J1 visa is not subject to section 212(E) two year rule meaning I do not have to return home for 2 years after my visa expires.

I met my (27f) Fiance (21m) online in October of 2023, we officially entered a relationship in February of 2024. We spent basically every waking moment together on call playing games watching movies/tv shows or just being in each other's company. We were completely long distance NZ -> USA until I came over here for my work visa. As unconventional as it is my Fiance did propose to me before I had come over and once I was here he presented me with a ring which I adore. It was always my intention to return home. I have flights booked and everything and have all my family back there, however that has very much changed now that I can be with my Fiance in person. 

We had planned to get married in December and still intend to, both of us are very happy just having a courthouse wedding and enjoying a day in the snow together. I have a couple family members visiting in January who we could have a small belated celebration with. As far as evidence of a legitimate relationship we have tens of thousands of messages between the two of us as well as video clips taken when we were in calls together. We also have venmo and paypal receipts from gifting to each other. We have also been living together in my apartment since mid June but it is just me on the lease so we do not have any documented proof of that. His father drove him here from his state and can vouch for him living here and meeting me. It may not be relevant but for the purpose of having all information available I want to mention that his father is retired military. We also plan to put him on my bank account and get a credit card together, 

A couple concerns I have about this process is the amount of evidence we have, I have no idea if screenshots of messages and clips of us playing games and watching things together are enough. I was also told by my employer that I cannot work for them while I apply for a green card. I may have misinterpreted what the person I asked was saying but I have asked multiple times and they say the same thing. And if this is true from what I’ve read, if I start the application process after my employment ends I won’t have enough time as I have seen it can take multiple months. Unless you are allowed to stay in the US while applying? 

I was also wondering if there was an alternative that would be recommended besides a marriage green card, that would allow me to stay in the states. My employer has offered the chance to extend my visa and come back for another year, and seek my own accommodation which would allow for me to put my Fiance on the lease but they have said I would have to return to NZ even if it was just for a month which I would like to avoid as it would leave my Fiance alone with our cat and no place to live, he does not currently work. 

I hope I haven’t missed anything too major out, I would love any advice or recommendations as it’s causing me a lot of stress and anxiety which I am already very prone to.