Loc: Ozark Mountains in SW Missouri
It sounds like your fears may be exaggerated a bit, but that's pretty common. Like anything else, you'll acclimate with time and your fears will naturally become more in tune with the realities of your experiences.
I've never seen a cougar up close, and only maybe seen a glance of one from a distance. I have only seen two black bears in all my years of hiking on and off trail. Both times they were doing their best to avoid me.
One of those times was a very close encounter, I stuck my head into a spot where the bear was hiding. I backed away slowly without a sound, put my pack back on and left. The bear stayed put.
I guess the best advice I can give you is to always give the critter a way out. If you run into one and it feels boxed in it will attack, but if you back off and give it an escape route it will use it.
If you stumble onto their den a critter will defend it, so again, back off. Don't make noise or act aggressive, just back off. In most cases they will stay put. I have a son and daughter who met up with a cougar at her den. She was clear in telling them to leave. They backed off and she stayed put.
Don't make the critters mad. I've gotten chased off by a groundhog whose den I was messing with. It gave me plenty of time to back off but I didn't. That made it mad and it started coming after me looking really mean so I ran as fast as I could screaming for help from God. The groundhog didn't keep chasing me though, it stopped just a little ways from it's front door and yelled obscenities at me. I had a vision when I finally stopped running. God was laughing in it. That's all there was to it.
So you have to learn from your mistakes, both big and small, to get comfortable out there, and you have to be out there to do that. Over the years I've learned not to poke my hiking stick into critter dens, not to stick my head into every crack in a bluff, always give a critter an easy escape route, don't pee on a moose, and God has a sense of humor.
I've also learned that it's me I have to fear the most out there when I'm hiking solo. Resisting the urge to do something stupid can be a challenge and it's really your biggest danger out there. I still haven't mastered that yet, but trying is part of the fun.
I'll also offer that you start off easy with your solo hikes. Do some day hikes in familiar places. Plan an overnighter there and make it someplace nice but still easy to bail out on. Then do a couple nights there. Find a park or an area and explore it a little at a time until you know it all very well and then spend a few nights or a week rambling around there.
A few of those and you'll be comfortable out there and after that you can go explore other areas the same way and enjoy it more because your fears will be more in proportion to the realities.