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It isn't really 'set in stone' what fingering you choose - it depends on how comfortable it is for you.
Yes, for the chord of C major (left hand), notes C E G usually use 5 3 1, (rule is miss a note = miss a finger) but sometimes 4 2 1 is used instead & more comfortable. Trouble is with some adult beginners, your hands have developed differently throughout your age, where a young kid can easily have their hands & fingers 'moulded' - as they know no different.

If you are having difficulty in moving from one chord to the other - which is usually the case when you have to keep a melody going, perhaps try a different chord position (called an inversion) for the chord of F. The notes do not have to be in 'root position' (FAC), but can be C F A, so in moving from the chord of C - C E G, to the chord of F - C F A, keep your little finger in C, and use fingers 5 2 1 for C F A. The chord of F will sound slightly different, but its still the chord of F.

This can also be done for the chord of G - B D G - using fingers 5 3 1. It means also that you have less distance to travel to each chord, instead of your left hand flying around!! Best to allow your fingers to move the "nearest available note" - for simple left hand harmony chords.

The general rule for 'ad lib' left hand chords is that if 2 chords share the same letter name, then try to keep that note in the same part for both chords.
C E G = Chord of C. F A C = Chord of F. Note in common = C
So turn around the chord of F to keep the C in the same part
C E G then to C F A is much smoother than C E G to F A C

Hope that helps as a start (but your piano teacher can help you further with this).....

If your chosen fingering works for you, and it still enables you to glide quickly and have smooth transitions to your next section then stick with this.
I'd try and work on some alternative fingering though in future pieces, and see if you can find a way that works for you to use your thumb.

You're up late, PL; are you annoying the neighbours with a spot of late night practising...?

G major chord then E minor. E minor's my favourite key!
It's good that you're exploring the piano, not just slavishly plugging away at the same pieces, PL. Have fun!

Play the chord...then just relax. No point doing otherwise; whatever you do to the key after you've pressed it won't make any difference, so you might as well hang loose, man. In the terrible old days when I taught I used to tell the brats to flip my hands up away from the keys at any time; if I weren't relaxed they wouldn't be able to do it. When I did the same to them, some of them were so rigid - fear, probably - that I wasn't able to budge their arms! Relax the whole arm and shoulder after you've played, do nothing else - and remember what that feels like.Then just play one octave of a scale, both hands, stopping for ages on each note and concentrate on keeping that feeling of relaxation. Mark will have some good tips for you. The more confident you get, the more relaxed you become - I'm utterly rigid when I play the guitar because I'm learning (I also can't play without sticking my tongue out...) but if I ever get more confident I might relax a bit!