I've tried a few different techniques of dyeing my facial hair, cause mine is similar to yours: dark on chin etc, but lots of blonde in my mustache. Here'e how different things worked for me:
Regular hairdye: stick to skin as well (for a few days after) but stuck really well to the hairs. About a week or two later though I had visible blonde roots so had to re-dye it again. Also stings like hell on earth on my sensitive face skin, especially mustache area. I was praying to the gods for my sweet release during those excruciating 30 minutes I left it on. Despite those issues, this became my fave method, cause it looked the best and lasted the longest. I used the shades black and dark brown, depending on if I was dyeing my head hair black or keeping it closer to my natural dark brown.
Hairdye intended for men's facial hair: doesn't stick to the skin and also doesn't sting. I've only used the black shade of it though, and I've heard the lighter shades come out way darker than it says on the box. It also didn't stay looking good very long, barely a week, before I had re-do it. I bought this (called "Just for men") online from a webshop that sells beard care and barber stuff in general.
Dye for eye lashes and eyebrows: very similar to the beard dye. This I bought at a regular beauty store that sells makeup, hair styling products, nail polish, fake lashes, etc, you know. It didn't stain the skin, looked alright, had to re-do after about a week.
Regular mascara: difficult to avoid lumps and looked weird no matter how I tried to apply it. Also felt very stiff whenever I was smiling. Only stays for 1 day though and washes off easily. But also maybe my mascara sucks.
Also if your mustache hairs are really thin it might look like the dye didn't stick, cause very thin hairs just look lighter even when they're not. My blonde hairs are really thick and coarse so dyeing them black (or dark brown) made a huge difference. They seem to be terminal hairs as far as I can tell, but just stayed blonde, for whatever reason. I've also thought maybe I'm going grey already, but unlikely considering I got those hairs around age 26. Then more likely my beard just ended up being two-coloured, with light blonde and dark brown, which does happen for some cis men too. I'm a calico!
These days I don't dye my beard though, cause I only let it grow to a stubble before I shave it again. But I like the stubble so sometimes I maintain that for a while. Despite detransitioning I still like my facial hair, and feel strangely proud of it despite seeing myself as a woman, so I really don't mind talking about it.
