Sounds like you solved your own problem. I think it's great that you are asking the questions. I taught myself guitar and music theory just in the last couple of years, so I know there are lots of questions to ask! And sometimes people who understand this stuff already don't realize how mysterious it is to those that don't, so as an educator and a latecomer to music theory, I can't resist at least trying to answer simply, as maybe it will help someone else, so at the risk of being too simple...
It does sound like you are talking about "transposing" the song to a different "key."
But it's not clear if you (or some other beginner reading this later) realize that, if you can't sing something the way it's written, so you want to change one of the notes or chords, what you actually want to do, most of the time, is change all the notes or chords, otherwise you'd be singing/playing off key. There are times people just change the melody, but they would usually choose a more sing-able note that's still in the chord and just play the same chord.
So, if you see a D, but it's too low to sing, and you want to sing a D#, up a half step, then, most of the time, you want to change ALL the notes up a half step. Which is the same as transposing the key, and the same thing the capo does, if you put it up just one fret.
You don't have to use a capo, of course, you can just play the chords that are in the different key, but then you change all the shapes and sometimes the voicing of the song, which is also fine if that's what you want, but a little harder for a beginner.
You describe sometimes the song sounding way off. It made me think of my own experience with this and maybe this will make it clearer. For example, there's a song written in Am. I can sing the song pretty well, but it sounds awful if I try to play Am while singing it. I capo up and down for a while trying to find the key I'm singing, but it's hard for me to do that by ear and it still sounds... not good. So I check using Melodyne to see what key I'm singing in and it's Fm. No reasonable amount of capoing is going to take an Am shape to an Fm sound in standard tuning.
I can do a few things. I can play the shapes for the key of Em and capo up to Fm. Or, if I want to keep the same shapes as the Am, I can play my baritone guitar for which the Am shape sounds an Em, and capo up to Fm (this is why I bought the baritone - it's easier for me to sing with). Or, I can simply play the shapes associated with Fm in standard tuning. But no matter how I do it, I'm changing the sound of all the chords in the song from the ones in the key of Am: Am Bdim Cmaj Dmaj Em Fm Gmaj to the ones in the key of Fm: Fm Gdim Abmaj Bbmaj Cm Dbm Ebmaj. This is useful for seeing which chords and notes go together:
http://scottdavies.net/chords_and_scales/music.html
It's also not clear that you realize that these chords come in predictable sets, most of the time. (I keep saying most of the time only because there are many varieties of music that don't follow the "rules" of most rock/pop/folk/etc of the western world).
Understanding those predictable sets comes from understanding the fundamentals of music theory. Some basic theory will also help you understand the chord embellishments and variations you see like Dmaj7 and D6 and such. I took an online course in it to figure it out, but there are lots of ways to learn. It's definitely worth knowing in my opinion, and it helped me tons.