I caution you against getting into the 'intro' trap.
When I first started playing after a few months I was where you are at and I agree, it can be frustrating and demoralizing. I got together with a guy I worked with who said he'd been playing for years. Right out of the gate the guy blew me away playing these cool, intricate intros to some great songs that I loved. I was so impressed. But then I realized that was all he could do! They guy could not play ANY songs beyond the intros. His 'talents' that he had developed were completely useless.
Learn the whole song. Sure, the verses are repetitive and redundant but they are part of the song. That repetitive nature is exactly what you need right now to develop muscle memory. Find easy, three chord songs you love and play AND sing them over and over. Even if you can't sing all that well it will still help you get smoother with your chord transitions. Don't worry that you are hesitating between changes. That is part of the process. It's all about muscle memory and it will come.
That was how I learned. I would learn three chords and then go find songs I knew and loved that used only those chords. Then I would play AND sing them over and over and over again. Eventually they started getting smooth. Then I would add another song that would have a fourth chord and learn that chord and then that song and so on and so forth. Just as soon as you can start to play a few songs start to finish, find yourself someone to play with with similar abilities (or slightly better) and taste in music and form a little band.
That's when your abilities really take off. At least, that's how it worked for me! In my case I dragooned my wife into being my lead singer and we started playing together. A few years in we started going to open mics. A few years after that we started opening or guesting for local bands. A few years after that we started getting gigs. And now about a decade later (in total from the beginning) we are booking over a hundred gigs a year and having the time of our lives in our retirement still playing easy and fun songs.
As far as how many songs to learn at once, I'd say as many as you can find that have the chords you can play or are working on but just make sure to practice them ALL THE WAY THROUGH!
As for song suggestions, that depends entirely on songs you like and how many and which chords you know! Go check out Party Marty on YouTube. He gives great demos of easy-to-learn, simple cowboy chord songs. He even breaks them down into the number of chords per song! Here's a direct link:
https://www.youtube.com/channel/UCOY...fXcg/playlists