Outcome : my visa got validated!
As per everyone's recommendations on this forum I went for option 1 : enter the USA before the expiry date on my visa.
Here is what happened, in case it can help someone else in the future.
I travelled with my husband (USC) and presented ourselves together at the Border counter.
We entered in NYC, at JFK airport. This is not the location or state where we intend to establish domicile and all our documentation in our visa process stated San Diego.
We had a return journey booked for 3 days later.
We were advised to join the queue for tourists and "all visas". The wait was 1h30.
The agent asked me if it was my first entry, to which I replied yes.
He said I would need to go to "secondary".
He asked me for the adress to send the Green Card to. I gave him a document showing the same adress I had given at the Embassy interview for this purpose, which is of a family member in NYC (as we don't have yet a permanent adress in the US).
He asked me to confirm in which city is this adress.
He asked me to confirm this is not in San Diego.
(At this point we were extremely stressed and were bracing ourselves for a refusal.)
He closed his counter and asked my husband to proceed past the Border area and wait on the other side.
At this point he was kind and reassured us saying we would see each other soon. (He saw we said goodbye and were clearly very stressed!)
He escorted me to the "secondary" room less than a minute away.
He then reassured me and another man that no one is going to send us away.
He asked me to take a seat in the waiting area, he put my passport in a folder and left it for his colleagues to pick up.
Another agent picked my document, entered some details on his computer and called my name to his desk.
He handed me my passport and said "you're done".
And that's it!
It was a very stressful experience (in our heads) but in reality we got asked no questions. I had a medical mention in my Visa "Class B req attention of USPHS at POE" but this wasn't mentionned.
When talking through the chain of events afterwards we realised that the first agent had already stamped and signed my passport before heading to "secondary".
Tips :
-I would recommend anyone to have their document ready and at hand shall they be required. A lady in front of us, also Visa holder, had a hard time digging up a document in her suitcase, in front of many, and it was an extremely stressful situation for her.
-Do not forget your suitcase at "secondary" because you are so overwelmed and relieved to have your visa! It happens to be very hard to retrieve after!
Thank you to everyone who helped us on this topic @pushbrk @Crazy Cat @Boiler @Dashinka @powerpuff @Mike E and @etrangais !
We are happy and very much looking forward to our move to the US!