There's not really much in the way of browser-based furry games that aren't a) pornographic in nature or b) directed towards children.
The only option I can offer that isn't pornographic and is at least somewhat related to furry is Transformice, which is a fun little platformer with basically infinite content through its level creator, but unfortunately its userbase is shambles at this point and at least when I played it was a lot of teenage girls, hitting a big crossover on the Warrior Cats fandom, or Brasilians that don't have money to play real games.
If you're looking for something pornographic, Dormaus Valley can while away a couple of hours, but it is entirely based around transformation. Paraphore can take some good time if you intend to pay attention to the in-depth story, but it also involves basically every fetish in existence except vore (maybe still - I never saw it of what I played). The overall story is kind of neat, I guess. You also have your heavy hitters of CoC and TiTS.
If you have a decently operable smartphone, you could get into Tokyo Afterschool Summoners for a good gacha game kick that includes furries. Personally I find it an unfulfilling game, but I play tons of Granblue Fantasy which unfortunately is very lacking in both furries and husbandos.
Maybe you can spend time learning how to make the browser-based furry game you want to play.