Personally, I would recommend that you learn the minimum amount of music theory to be able to write down your music in the kind of form shown below (at least as a starting point; you can develop into writing a full score later, if you want/need to):
1 3 5 5 3 5 5 3 5 5 3 6 6 5
You're strutt-in' in-to town like you're sling-in' a gun, just a
The top row shows the harmony (chords), the next row down shows the melody (notes). If you're writing a song, then you can also write the lyrics below that to show what syllable goes with what note. Otherwise, the melody will be played by some lead instrument instead of being sung. In my notation above, I put square brackets around a bar's (or measure's) worth of music, and the music is in 4/4 unless I state otherwise, so the I chord is played for two beats and then the V for 2 beats to make a bar.
You can probably learn enough to write in the above format if you just study this: http://voices.azurewebsites.net/elements/notes.htm
. Study "Notes and major scales", (you might be able to skip "Intervals" and even "Minor scales") and "Chords". That should probably be enough.
"lucky enough to be able to imagine my own songs in my head." IMHO, whether that's lucky or not probably depends on how satisfied you are with your work. If you're satisfied, then you're extremely lucky. If you're not, then I'd advise looking to expand your composition toolkit.
Generally speaking, a composer uses three mechanisms (three tools, or processes), and IMHO they're all necessary (that is, none is sufficient on its own), but they're each important in different degrees:
1. Inspiration. This is what you refer to as imagining songs in your own head. You either make songs up in your imagination while you're awake, or you wake up after sleep, still remembering them. I'd argue that this is the most important mechanism of the three. While it's far from ideal to only have one, if you do only have one then this is the one to have.
2. Prediction. This is where your knowledge of music theory, and how melody and harmony interact like characters in a novel, and your knowledge of how humans respond to suspense, comes into play. You use it to guide your composition, like a good city guide using their experience, instincts, and knowledge to direct their feet to the places that are most likely to yield interest. It's not guaranteed, it's just a good guide, like a dog following a scent; and there are scents going off in all directions and dimensions, always (if you can detect them). This is probably the next most important mechanims.
3. Discovery. This is the teenage guitarist in their bedroom randomly strumming chords until something "sounds good". Most amateur composers never develop beyond this mechanism into the two above. However, if you can bring all three mechanisms to bear, then discovery can still have a decent place in your toolkit, and can have its uses for sure.
Basically, IMHO, you can use the synergy that arises from using all these three mechanisms together to compose.