My main problem is that, whatever I play always ends up sounding like someone else's song
Within a genre, many, many songs use similar basslines. Listen to old school country - do they have drastically different bass lines? Listen to classic rock - while there is a great deal of variety in bass lines in general, there are many, many songs that use similar stuff. There are reasons for this.
First, they work. Because they worked once, they continued to get copied or mimicked by future players.
Second, there just aren't a ton of options available for single note lines - that sound good.
I'm not saying it isn't possible to create new types of lines - it's just that the old ones work - not broke so don't fix it.
Since you can create stuff for other parts, you have the ear to do it for bass as well. My advice is this:
1. Listen to as much music that is similar to the style you're working in. But also listen to many other styles - new bass line styles tend to come out of blending two genres (playing a reggae bass line under a c/w song for example. (I've never done this, but I bet someone has.
Take one song, get a feel for what the guy does and then start trying to mimic that line line. Then try to create a slightly different line. If you are really lost, start by playing just root notes. You just wrote a bass line. Then start adding to that maybe doing root/fifths. Then try to build on that line for that same song. Keep at it for a while - you'll probably find that you can create several passable basslines for any given song. Pick another song and repeat. (I tend to write my own bass lines in this manner for nearly every song I play.) Whenever its applicable, I try to make the bassline something one would want to hum. If it doesn't apply to do that, then basic Root-V variations and arpeggios are likely good starting points.
2. Do get some basic theory under your belt"
- Learn and understand your major, minor and pentatonic scales (being able to name the notes is beneficial.)
- Learn and understand your arpeggios.
- Learn and understand the circle of fifths
- Learn some of the basic common chord progressions (I IV V, II V I, or what ever is applicable to the style of music you're working in. (If you don't know what ("I IV V" means then learn that too.)
3. Marry your theory with your listening. See if you can understand
why the notes being played sound good.
It really does just take some time to immerse yourself into this but once things click for you, it will become second nature.