Some jobs don't require that you be married to get added to the insurance. Your partner should contact HR to see how it works ASAP. They do typically require that you live together, but some places are more lax than others. They should also look at the paperwork needed and how long it takes to kick in, and also how much it costs per month to add you (I've heard a lot of dependent coverage increased a lot per month).
Until you are on your husband's policy, you might be able to qualify for short-term health insurance. You'll have to call to see if you qualify. I used this insurance when I took time off between 2 jobs and it was like 80 dollars a month. Even though travel insurance might not work, I would still do some research about it, because a quick google search left me a bit confused and it might be an option at least until you get married. My dad had travel insurance and they never asked him about his visa or anything, and he did have to go to the doctor once.
If you take medication, I would recommend you talk to your current doctor to be able to buy and bring your medication. Between the insurance kicking in, getting an appointment, and getting the medication, it can be a couple of months. Doctor's appointments can be difficult depending on the area, so even if you don't take medication, I recommend you make an appointment as soon as you get insurance.