I’ve got a few thoughts here and will try to address everything:
1.) It is very common for men and women to speak in a different tessitura than they sing. I know men that speak high and sing low and vice versa.
2.) Typical tenor and baritone range differ by about a perfect forth, so the trouble you’re likely to have will be connecting the low end of the baritone range to the low end of your tenor range.
3.) Speaking with vocal fry is incredibly tiring for your voice. See if you make yourself more aware of when you’re doing it and correct it. Usually for me it just means I need to speak a little louder.
4.) There’s nothing “girly” about using falsetto: boys and girls have the exact same voices until puberty.
5.) You didn’t say how old you are now, but depending on your singing history (and I’m guessing you’ve been singing for a little while since you’re comfortable using the terminology and you’re not classically trained), but it could be possible that you developed a strong, coordinated sound before puberty, and then when your voice changed, you’ve mainted the pre-puberty voice and have never properly learned how to access your new lower range. Men need a lot of help in this way.
6.) Even though you talked about it like a negative thing, instinctively, you’ve been doing a good thing: bringing your chest voice up and bringing your head voice down. Vocalists do this to create a mixed, coordinated sound that help to create a seamless instrument. Use lots of air; be careful that you’re not belting, but “sounding like an opera singer” is actually a good thing; it means that you’re doing a lot of things right and achieving nice resonance, rich timbre, lots of air, and a coordinated sound, among other things. Everyone thinks that if you don’t sing with a breathy pop sound then you’re immediately singing opera (thanks “Greatest Showman”....) but it’s sinoly false information / thinking.
7.) You should definitely consider voice lessons. Especially with someone with a background in vocal pedagogy, that will help you be able to learn the mechanics of your vocal mechanism.
8.) Irrespective of genre, everyone has issues with their passagio.
9.) Singers often get sucked into “yeah but this is my vocal fach”. A wrong classification can tailspin singers for years. Also, you’re voice won’t fully mature until your mid-30’s, so your fach will likely change. Go where you sound good / feel good - you and your voice should feel good; don’t worry about what you’re “supposed” to be doing and strain yourself unnecessarily.
Hope this helps.