Frustrated!

I recommend that you read the 'Basics' section of the manual carefully.
In most cases the input mode 'Step time (default)' is the right one.

For example, entering chords with notes of the same duration is quite simple: first set the lowest note, e.g. C, and then use Shift E and Shift G to write the C chord. Only if you want to write chords with notes of different durations, you need the voices 2, or even the voices 3 and 4.
Changing note durations is a bit more difficult, because a 4/4 time signature has only four beats and before you make one of the notes longer, you have to make another one shorter. Otherwise, all that's really left is to cut and paste in the right place.

"Does anybody have any tips for dealing with this??"
1. Always work from left to right. Musescore will maintain the correct number of beats in a measure by adjusting the duration of the remaining rests.

2. If you find that you have entered a short section in the wrong place, you can move the section:

- select the range to be moved
- cut the section into the clipboard
- position the cursor on the new start position (rest or note)
- paste the section

You wrote:
Why can't you just drag a note from (say) beat 2 back to the rest on beat 1 instead of deleting it from beat 2 & re-inserting in beat 1?

Okay, so you drag a note from beat 2 and "get rid of " the rest on beat 1. Next, you hit the play button. What happens on beat 2? There is nothing there!!!!

Were MuseScore strictly a graphics (no playback) program with a "toolbox" full of musical symbols that one could place anywhere - and drag/delete as necessary to make edits - we all would need to be skilled music notation typesetters. After all, one might "drag" 4 quarter notes and space them closer than, say, 4 sixteenth notes. Multiple accidentals which may need to be "stacked" for a chord - especially for a chord with noteheads on each side of the stem (2nd interval) - would be difficult to space and align correctly.
Such freedom might be wonderful for individuals to do as they please, but terrible for the playback engine, or any musician expected to read from and play such a score - especially because someone could conceivably "drag" a note into the title bar!

There needs to be at least some framework of rules which allow for multiple users to achieve consistent results according to generally accepted music notation principles.
Keeping that in mind, you also wrote:
...when I change the value of a note, everything after that point shifts, sometimes pushing notes into the next measure.
Hmm...
"Everything after that point" does not shift. If your score had 1,000 measures you'd be screwed if everything moved. The rule to keep in mind is that pasting (or changing to) a note of a greater duration will consume only what additional time is actually needed - no more, no less.
For example, if you change to a whole note on the final beat of a measure, the whole note will get tied over into a following measure. It will overwrite any rests/notes as needed.

Conversely, changing an existing note to a lesser duration will introduce rests to fill up the timing.
For example, if you change a half note into a quarter note it will generate a quarter rest to fill the truncated (half note/2 beats) duration. That way, the 1,000 measures which follow won't all get moved up in time to fill that gap which the quarter rest now occupies. Remember, the playback engine and any musician reading the score needs to know the precise timing of notes and rests.

You wrote:
Isn't there some way to prevent errors from spilling over into following measures? Or turn off the automatic stuff within the measure until I get it fixed up manually?

You might consider 'Insert' mode for such occasions:
https://musescore.org/en/handbook/4/alternative-note-input-methods#inse…
"This note input method allows you to insert and delete notes and rests within measures, automatically shifting subsequent music forward and backward within the measure. The measure duration is automatically updated as you go."
The caveat for using Insert mode is that after use, be sure that measure duration is correct relative to the score's time signature.

There is also a plugin:
https://musescore.org/en/project/duration-editor
which works for MuseScore 3.

You wrote:
I am slowly learning this program,...

Your contributor profile shows you active for 10 years... :-0
Don't give up now... ;-)

Yes, learning any sufficiently complex system takes time and can be frustrating. But, it seems you might be confused about a couple of things, but it isn't totally clear what. One thing I can say for sure: changing a duration does not move any other notes. Every note stays exactly on the beat you entered it onto. Unelss of course that beat is now part of a previous note you just lengthened, in which case the bare minimum of changes are made to ensure nothing isn't moved that you didn't specifically just ask to move. Also, changes in one voice never affect any other voice.

So, if you continue to have trouble after reading the replies here, please attach your score, and give precise steps to reproduce the problem you are experiencing. Then we can understand and assist better