Everyone has given you great advice. And to be honest, you do a very nice job with the song.
My only suggestion going forward is to try and incorporate other fingers on the right hand if you are able. I imagine it will be a slow process.
Trying to play every note and chord with your thumb is really tough. I’m not sure I could play fluently with that technique.
This will really smooth things out since it becomes instinctive using the index, middle, and ring finger and not moving your thumb all over the strings. It will also give you some variety in sometimes plucking multiply strings instead of a downward strum.
Different strokes (or strums) for different folks. I am a chord melody player and use only my thumb. I prefer it over the different dexterity required to use other fingers. I am currently using a thumb pick to play.
Three of the video instructors I follow utilize mostly thumb for chord melody lessons. They are Stu Fuchs, Matt Dahlberg, and James Hill.
So you can try both styles. But switching to using the thumb and two or three fingers will likely set you back in trying to play songs fluently while you learn this new technique.
Just to clarify... I think there has been a bit of confusion. I'm not the one in the Moon River video. I wish I was! I just posted it as an example of the kind of arrangement I usually practice.
But, yes, I play mostly with my thumb. I started learning chord melody with James Hill 'Ukulele Way' course and he recommends that before start using the other fingers. And since the biggest difficulties come from my left hand, I want to become more fluent with it before complicate things also with my right hand.
Only in the last few weeks I have started to use the other fingers, mostly to fight the monotony.
Interesting timing on this and I’m going to echo with what some others have said. Just last week I asked my instructor about practicing. He said there is a difference between practicing and playing. Practicing is focusing and replaying a part that is difficult for you, repeatedly. This is to help with the muscle memory and build your hand strength. No need to do this for incredibly long periods of time he even said 5 minutes would be a long time. Then there is playing, and that is playing a song fully from start to finish even with the errors. Ultimately you need to utilize both of these to improve and continue to have fun.
Another thing I’ve learned that will help with this is playing a song you know well and enjoy listening to. For example, let’s say you already listen to Jack Johnson’s Any Wonder regularly and you also happen to find a transposition of the lyrics and chords for ukulele. You will already be able to play this better than you normally would because the music is familiar to you. I would highly advise to then play along with the actual song when playing once you are comfortable with it. I think you would enjoy this and be surprised at the results.
Edit 7/26: I started learning Any Wonder on baritone uke this week
The dark side of playing a song you listen to a lot is the impact on your partner. I once asked my wife what's my favorite mode. She said with venom, "phrygian" and she was totally right. I guess I've played it so much that it had become a mantra in the household except that instead of leading to nirvana it was dragging the household to hell.