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I tried to look around for an answer to this problem but don't know how to ask the proper question in a search engine. I've been working on a song and realized recently that while I had been working on the song in the key of Em (E, F#, G, A, B, C, D) I had accidentally placed D# notes in several of my melodies. The confusing part is it sounds fine and I didn't even notice until I looked closer. What's more, if I change them to D it sounds off key and if I change them to E it just sounds wrong for what the melody is supposed to be doing. I don't have much music theory knowledge but this confused what I thought I knew. Am I misunderstanding something here? Any help and clarification would be appreciated.

Your confusion is understandable because you have the choice of using one, or a combination, of three minor scales: the natural minor, the harmonic minor or the melodic minor. In using a D# you have strayed from the natural minor scale to the melodic minor scale, and this scale has worked for you.

The natural minor scale flattens the 3rd, 6th and 7th degrees of the parallel major scale. The harmonic minor scale flattens the 3rd and 6th. The melodic minor (ascending) only flattens the 3rd, but when descending it is identical to the natural minor, flattening the 3rd, 6th and 7th. The jazz melodic minor just flattens the 3rd, ascending and descending.

These scales are all legitimate minor scales (there are others too). What's known as the Leading Tone, in this case D#, has a strong pull towards the tonic and it is often used in popular songs. At present you seem to prefer that semitone sound between D# and E, and that's fine.

You'll most likely find that the chord used where D♯ fits better is either B major or B7. That has the D♯ in it. The D ♮ will fit in other places, notably when going from an Em chord to an Am. It's the reason centuries ago that the natural minor scale morphed into the harmonic minor, with a raised leading note ( here, the D♯), and later, because there was then a great big jump of a tone ane a half created, the 6th note of that natural minor got sharpened too, to remove a big jump.

All the minor scales have the same first five notes, but the melodic minor (classical) has the same 6 and 7 as the parallel major scale.

D# makes a nice lower neighbor to E. Likewise, in E-minor, the dominant (B) often has its third raised at cadence points. Minor keys (at least in classical theory) have two mutable notes; scale steps 6 and 7 may be raised to make voice leading smoother or just because it sounds good.

I tried to look around for an answer to this problem but don't know how to ask the proper question in a search engine.

So you are writing a piece in e-minor and obviously you know what is e-minor, you know the scale of e-minor is and you know that there are D and D#.

I had accidentally placed D# notes in several of my melodies.

Why sounds D# more harmonic?

As you will see you are not the first that The human mind already in earlier times of music history obviouly wanted to have a tension to the final note (the root) and so they constructed a leading tone by augmenting the 7th degree.

This was the "invention" of the harmonic minor scale.

What's more, if I change them to D it sounds off key and if I change them to E it just sounds wrong for what the melody is supposed to be doing.

By augmenting of the seventh degree was a gap of 1+1/2 tone between the 6th and the 7th degree: in e-minor between C-D#.

So to become a more "natural" ending and easier to sing (for instruments it didn't matter) but also for listening they augmented the 6th degree too. This was only needed when the melody was leading upward to the upper octave. In a downward melodic formula the leading tone was not needed and so

the the melodic minor scale had been developped.

You are free to use all kind of the 3 scales at your "gusto".

Just looking for e-minor:

here is explained the e-minor (and all minor scales have 3 variant models)

https://en.wikipedia.org/wiki/E_minor

I don't have much music theory knowledge but this confused what I thought I knew. Am I misunderstanding something here?

If you want to continue composing my advice is to read first the basics of music theory (scales, chords, harmony).

and again:

I tried to look around for an answer to this problem but don't know how to ask the proper question in a search engine.

I was looking up developpement of melodic and harmonic minor and found this link:

https://study.com/academy/lesson/harmonic-minor-scale-formula-modes-quiz.html