I am so paranoid about SATB harmonisation, because there are so many rules, and I feel overwhelmed. I literally feel like I have a hundred hidden mistakes in this question.
- A minor is a problematic choice for your realization, for the reasons you've discovered. Were A minor the intended key, and unless this is a relatively advanced exercise, I would expect all of the given Gs to be G#s. Minor V is unusual in a beginning exercise, as is the prominent use of III chords.
The most important factor in answering your question is whether or not the key signatures are given, or whether part of the exercise is determining the key.
My feeling (with an assist from @MattL) is that the strongest realization would be in G major, ending on a half cadence (D major -- the V chord of G major). Nearly all of the melody notes are part of the G major triad, with the non-triad-tones (A) in weak rhythmic positions -- clearly passing tones.
If you're supposed to use the given key signature, then I would realize this exercise in C major, ending on a deceptive cadence (A minor). That's because of the G naturals. Were the excerpt to be in A minor, I would expect G#s (unless this is an advanced exercise where minor V is expected). However, one would still need a G# in the second-to-last chord, risking a cross-relation with the G natural at the beginning of the measure.
Is this really presented with no key signature?
That is the most confusing part about it. The melody so obviously is in
G major (key signature of one sharp) but this has a key signature of zero sharps/flats.
The reason the key signature matters isn't a nit-picky gripe, it has bearing on the strategy for harmonizing, which should be done with chords that make sense within the key!
You should already have some harmonization procedure, but here are a few strategies you can consider. Ideally you want to understand the harmonic implications of the tones of the melody within the key and first fill in a bass that supports those harmonies.
Try a "reduction" to get the main harmonic events...
...that can be fleshed out a bit with some of the other strong tones which are also from the outlined triad which we will consider the tonic...
...notice above how
^2 are first recognized in terms of their harmonic/tonal role within the key, this is important. For example, consider
B4. In some harmonization methods you will be told you can harmonize it with triads
vi. That's true, but in tonal music tonic/dominant harmony is the primary focus. In that regard
^3 is a member of the tonic chord. Similarly
^2 could be part of a
ii triad, but in terms of tonic/dominant it is a member of
Look for melodic/voice leading formulas. This is probably only going to come from experience. I noticed some "voice exchanges." That's when the root and third of a chord "exchange" or alternate in a part. An common harmonization of an exchange is just the reversal of the move in the harmonizing part...
...that give a nearly complete bass. Fill in the inner voices...
...that may be a boring harmonization, but the original tunes only outlines a tonic chord. You could move around some of the tones of the inner voices to give them a bit more interest, but I don't think it will matter much. You could harmonize it some other way than tonic/dominant, but I think the point of the assignment is use basic harmony.
The important thing is to understand the tonality of the original melody and first add a good bass. Adding the bass alone should make the harmony clear. It will be only two parts, but the implied chords should be clear. Filling in the inner voices is then fairly straight forward.