Professional facial hair removal isn't cheap, and insurance doesn't cover it (or at least none that I've heard of do). But it's worth it. It's so worth it. It does take a while, even with dark hair and light skin (the "ideal" conditions for lasers which work off color contrast). Just make sure the professional is a professional. Removing facial hair is not like removing hair from other parts of the body, or even like removing facial hair from a hirsute ciswoman. There are a LOT of follicles, they are very close together, and they are not all active at the same time.
If it helps at all, here's my personal experience with professional hair removal:
I had my first three treatments done by an accepting, kind, and well-meaning woman who had no idea what she was doing (I was maybe her second trans customer) and who used an old laser. It hurt intensely, it didn't have any visible effect, and it left my skin looking raw and sunburned for weeks. I've had my next seven done by skilled professionals who have been treating transwomen for years and who had new equipment. It barely hurts at all, the side effects clear up in a few days, and we're making progress. I'm over halfway there, but I still need many more treatments.
I've spent $1,335 so far (the first woman charged $125 per treatment, the professionals charge $160) and I will have spent over $2,000 by the time we're done. The professionals do treatments every 2-3 weeks, and the first woman wanted 4 weeks between treatments (which is really too long), so I've been at this for seven months so far. It'll be another 3-4 at least before I'm done.
Even after those ten treatments, I'm still shaving three times a day, but the hair is substantially thinned out. There are just a few places where I can clearly see hair after six hours or so of not shaving (mostly splotches on the upper lip and to the left and right of the chin), rather than having a nearly full goatee after that amount of time. I can still feel the sandpaperish texture of hair in those places after that much time has passed, but I can no longer use it to scratch itches on my hands and arms like I used to. I also no longer look for ways to hide the lower part of my face in public... except for when I'm on my way to a laser appointment wearing my beard of saran wrap and topical analgesic
EDIT: Oh, and I strongly recommend, if you see a professional for laser (I can't give advice about electrolysis), asking about how new their laser is. Places with newer lasers are likely to charge more, but it's almost definitely worth it. Ideally, you want a laser that also blows puffs of cool air as it works. That will prevent the heat from building on your skin, reducing the chance of burning you and reducing the pain. That lets them use higher power settings and/or treat a specific area longer in a single session.
Regarding laser vs electrolysis, the general rule that I've heard is go with laser as long as you can, then switch to electrolysis if necessary afterward. Laser is faster and cheaper, but color-dependent. Laser cannot get rid of blonde or red hairs, and it is less effective the darker your skin is.
Before a laser treatment, remove all makeup and shave your face as close as you can. The less hair that leads to a follicle, the less heat required to kill the follicle (or so they told me). But, be careful not to cut yourself or leave your face raw and red. I don't think they can treat areas with open wounds, and if you make your face redder, the laser will be less effective. Also, I think plucking before laser is bad - they need hair connecting the follicle to your skin to act as a fuse transmitting the laser's heat (I think).
Before electrolysis, you actually want your hair to grow out a bit, I hear, so it's easier for them to find the follicles. But, again, I don't have any first-hand experience here.
