I don't know about you all, but when I'm writing I feel like every other line needs to rhyme. If it doesn't, it bugs the hell out of me! It just doesn't sound right if it doesn't rhyme.

I feel like this holds me back from getting all that I want to say out onto paper because I can never find the right words. I've finished a few songs, but they always come out sounding like some sort of rap/poem. I'll listen to other songs for ideas, but then it feels like I'm almost taking too much from their song.

Anyone else feel the same way? Any suggestions?

By the way, I guess I can consider myself a "new" song writer. The best advice someone gave me was that the best lyrics come from emotional events in your life, like losing your girlfriend or when something exciting happens. This is so true, most of the time I cannot write unless my emotions are running wild. So on top of that, any other advice you'd like to share?
"I'd rather be hated for who I am, than loved for who I am not."

~ Kurt Cobain ~
I think you just have to immerse yourself in music that doesn't obey obvious rhyming schemes. Spend some time with some classics - Goffin & King were noted for their use of internal rhymes, for example, and Lennon & McCartney were hugely influenced by them.

Take a song or two and write some lyrics copying it's rhyme scheme - don't worry about melody, just focus on the relationship between meter and rhyme, just to get that into your head.
try using some less obvious rhymes, you know? slant rhymes can sound interesting. it'll still be a rhyme, but it won't be like Eminem rhyming me be we see in every other song.

also, to avoid that rap kind of sound, try working on your melodies. you can write almost every single song with an ABCB rhyme. if your melodies are all different (different phrasing, etc.), they won't sound the same.

there are a ton of different rhyme schemes out there, it's ridiculous. just put the rhyme where you feel it should go. if it sounds good, nobody will say "oh, every other line rhymes, blah blah blah."

my theory about everything: if your song sounds good as a whole, it's a good song. you don't always need the most complex music or the most poetic lyrics. it's music. it's about sound.
Rhymezone and even the humble thesaurus are your best friends when trying to find the right word for what you want to say. It sometimes takes a few searches to pin down the word you need, but with those two you will generally nail it.

As mentioned, perhaps different rhyming schemes can work. I am guessing you are stuck in the A-B-A-B format, for example, in the verses. I often like to expand on this with different chords to give the song a new flavour, for example A-B-A-B-C-C, then perhaps repeat this in a new verse.

The chord changes will make as much a difference to the feel of a verse as the rhyming pattern. Trial and error is the key.
I really dont rhyme, even though Im a poet. I somewhat do but I let the melody structure take precedence over the song. I never start a song without an inspired melody or baseline. Hardly ever with lyrics. I try to make them sound as best I can, but I'm having problems making my songs sound well formed.
^ that's true, nobody says you have to rhyme.
making unrhymed songs sound good is difficult - for me at least.

Mike Kinsella pulls it off beautifully.
I'd say definitely check out American Football and Owen, if you never have.