Kirsti, the relationship between androgens, oestrogen & systemic dryness seems to be complicated & from everything I've read the bottom line seems to be, as so often with women's health issues, that more research is needed. There are some interesting articles suggesting that androgens are important for the health of mucous membranes & glands involved in lubrication, as well as muscle function in the pelvic floor & of course sexual function, but it seems to be the
balance between oestrogen & androgens that may be the key to optimal function of the regulatory systems involved. To complicate matters further, there's some evidence that genetics also determine how individuals respond to oestrogen & androgens in this respect. It's pretty mind boggling but interesting if you are scientifically minded.
I've had a quick trawl through several bookmarked articles this afternoon but they are lengthy & complicated & I don't have time today to read them in full to refresh my memory, in order to say anything with much certainty or copy out relevant sections, but there are some links below you may like to look at at some point that may help you decide whether testosterone may be an avenue worth pursuing.
Yes, the most obvious improvement in my systemic dryness from supplementing with testosterone post-oophorectomy has been in the mouth. The dryness (all over) got markedly worse after losing ovaries, a major source of T production & I asked for a trial of T partly because Ophthalmology were getting more concerned about the worsening dry eye syndrome which was a worry, as well as being very uncomfortable & I suspected the reduction in androgens was responsible. I remembered years before BSO on first starting HRT with Evorel Conti containing the androgenic progestogen Norethisterone, that within days I'd noticed a marked increase in production of saliva. At that age I hadn't even realised my mouth was especially dry & was very surprised at this unexpected effect of HRT. When I went back on HRT a year after BSO it wasn't an androgenic regimen & my dryness didn't improve as much as I'd hoped so that's when I asked to start T & it has definitely made a positive difference. No guarantees, but it just might be the missing ingredient for you.
If you've had a full autoimmune screen that's good. Mine was also negative, yet the Sjogren's specialist Rheumatologist I saw still strongly suspected I had Sjogren's & wanted to do the gold standard diagnostic salivary gland biopsy because bloods aren't always reliable for Sjogren's. So if you are concerned this might be the cause of your dryness I think, as you say, it would be wise to get specialist opinion.
Anyway, here are some links & sorry I haven't had time today to pull out some meat for you & that some will seem similar, but if you like detail it might repay reading them closely if/when you have time.
https://www.sciencedirect.com/science/article/abs/pii/S2050052120300330https://www.sciencedirect.com/science/article/pii/S2050052118300465https://www.reviewofophthalmology.com/article/hormones-in-dry-eye-a-delicate-balancehttps://www.healio.com/news/ophthalmology/20120331/gender-and-androgen-imbalances-are-keys-to-dry-eyehttps://onlinelibrary.wiley.com/doi/full/10.1111/cxo.12147https://www.latimes.com/archives/la-xpm-1999-may-31-he-42741-story.htmlPlease PM me if you want more detail of my history.
Wx
P.S. forgot to say have you looked into Intrarosa/Prasterone (vaginal DHEA) for the VA? Probably a couple of years since I read up on it so I'm not up to date with any new developments, but if I remember rightly the thing that put me off was mention of a possible link to abnormal cervical smears. Don't quote me on that though, as my memory is not that reliable

so it'd be worth looking into for yourself.