I've been at my current job for around 7 and 1/2 months, and have been out to coworkers for almost that whole time. I do still get misgendered, although it's not constant. Some coworkers seem to struggle more than others, too. I find that I actually get more upset by the misgendering now that it happens less frequently, because I'm always more surprised when it happens. I also started to go by my new name in March or April, and have found that my coworkers have no trouble at all calling me by the new name, despite still struggling with my pronouns.
The oddest thing to me is when a coworker actually use my new name and then uses the wrong pronouns in the same sentence, because I had sort of thought that dropping my 'typically female' name and using an ambiguous one would remind my coworkers that I do not use 'she' and that I'm not a girl. Unfortunately this hasn't been the case.
In regards to your partner, I can relate to not wanting to have to correct people. It can be awkward, for both people. I have found that not correcting people can lead to people literally not even realising what they've said, however. I really dislike having to correct people and so I often get too nervous to do so, but when I do I usually do it in one of two ways.
1. I simply interject, although not in an impolite way, with just the word 'they' or 'them' straight after being misgendered.
or
2. I repeat the sentence with the correct pronouns. Eg. someone says 'I was speaking to Kolby and she told me about her game'. I would then say 'They told you about their game'. Again, I don't do it in an impolite or confrontational manner.
In the case of someone who still repeatedly misgenders you after being corrected several times, that can be a more difficult situation. One way to tackle this is to confide in someone who does get your pronouns right, and who you can trust, and ask them to keep an ear out and correct people who get it wrong. This can also be a good way to deal with being unable to correct people yourself. And peer pressure can work quite well; if someone keeps using the wrong pronouns while others are using the right ones, after a while they're probably going to feel awkward enough to try harder, as brutal as that might sound.
Also, depending on your partners changes while on T, it's probably going to get a bit awkward for people who misgender them after a while. Your partner shouldn't have to rely on these changes to be referred to in the correct way, however.
This has gotten long and I'm not sure how well I've phrased any of it, but I have one final point to make. When I got my new job, only one of my coworkers even knew what non binary meant (and she's my housemate who I've known for a few years now). Every single one of my coworkers has had to be informed of what my gender is and what it means. Even now, I'm 90% sure most of them don't really understand that I'm transgender and not 'a gender noncomforming woman who doesn't like to be referred to as one'. I'm hoping eventually they, and people in general, will understand better. But at the same time, it can be difficult to talk about, and there's always the fear that I'll come across as 'too forceful' or 'too angsty' just by trying to express that I struggle sometimes. What I mean to say is that it can be very difficult to work as a non binary person, and it can be even harder if even your coworkers don't get your pronouns right. I really hope that things will improve for your partner, and that my advice about correcting people, or any of this rambling, might be of some use.
