Autism

Especially as I am the first person who has been accepting of them going non-verbal.

I can only upvote this post and tell you how lucky they are to have you.

Can they write? Or is this a total shutdown of ability to communicate?

If it's the latter, depending on what is helpful for them, maybe just a hug, or something calming like some noise canceling earphones. For me, headphones can be a huge help when I'm overstimulated.

Otherwise just continue to be supportive, I think. They're lucky to have someone who cares enough to ask.

Going temporarily mute can be a really weird experience, and is something I imagine is very personal to the individual. When it happens to me, I usually end up wanting to be wrapped up in a blanket or thick hoodie in the dark in a soft, quiet place like my bed. Sometimes I'll stim with a toy or something, and sometimes I'll just kind of dissociate until I have the energy to speak again. If you haven't already heard of it, Stim Punks Foundation has some amazing content: (this links to their situational mutism page). Here's a great graphic from them that might help too. It's cut off but the full thing is on that page I linked and includes sources with further reading.

For general public interactions: Find a place that prints business cards and order some with something along the lines of "I stop being able to speak when overstimulated. Please give me some time. Sorry for the inconvenience." printed on them. Being able to pull one from a pocket instead of struggling to explain while unable to speak should make it less stressful.

Me and my partner then switch to texting on our phone or just asking question or formulating questions so they can answer with thumb up or thumb down

It can be useful to have an app like speech assistant or another AAC on a phone or tablet, or like me, it's on my phone, my partners phone, and my tablet, prepped with both the ability to type out my messages or prepared "cards" with words or phrases on them. I have periods of non-verbal or semi-verbal times, and it is really nice to have the quick ability to communicate. Yes, you can text, but if I am out on my own, I might not know everyone's phone number who I might need to communicate with. Imagine having a semi-verbal/non-verbal episode right as your trying to order coffee. (Don't be scared, be prepared, right?)

In my experience becoming non-verbal may also depend on who you're with. It might be that your partner goes non-verbal with other people first and you last (since you are the most accepting of this).

If you're able to speak for your partner in those situations (I suggest discussing how they should signal to you that this is needed, plus in what situations it would be appropriate) that can be a big help.

Also, as a couple of other commenters have mentioned, look into other forms of communication. Texting or email is often still possible if not too overwhelmed, so again discuss with your partner - perhaps there is a shorthand set of messages or emojis they can use when they're struggling to communicate that will allow them to convey what's happening and what they need.

Is this something that happens with others? Or only you and other people close to them?

text to speech. preferably a foss one like https://f-droid.org/en/packages/com.danefinlay.ttsutil/