Changing chords feels impossible because you are moving three fingers at once. So begin with changes where you barely have to move anything. There are more of them than you would think.
Am to C: one finger
Strum Am and C from the A string. Put your hand on Am: first finger on the 1st fret of the B string, second finger on the 2nd fret of the D string, third finger on the 2nd fret of the G string.
To get to C, move only your third finger, from the G string to the 3rd fret of the A string. The other two do not budge.
Go back and forth ten times without strumming. Watch your first two fingers. If they lift, even slightly, start again. They are your anchor.
Am to E: the same shape, one string over
Look at those two diagrams. E is the same shape as Am, moved over one string. All three fingers move, but the shape does not: keep your hand in it and shift the whole thing towards the ceiling. Strum E from the low E string.
Now put it in time
Strum four beats of Am, then four of C. Then four of Am, four of E. Your only job is to land on beat one. If you miss, keep going and catch the next one. A rough change is fine at this stage. Just keep going.
Why this matters
Do this for five minutes today.