"...if I'm a songwriter not singer and I only have lyrics..." Lyrics alone are not songs, which by definition must also have melodies, so you're not a songwriter but may be a lyricist. I say "may" because I've seen a lot of work that might pass as poetry but fail as lyrics. (To be fair, a lyric seldom functions as a poem.)
An experienced composer wouldn't be interested in working with novice lyricists unless they were obviously gifted. I suggest inquiring among musicians in your region for collaborators, particularly local songwriters with strong music composing skills. Advertise for "lyricist seeking composer", outlining the genres that are your principal interest.
Your first goal is to learn to work with a partner, which among other things requires that you both have flexibility in your crafts. You must be willing to edit lyric to fit the music, metrically and in other senses; the composer must be prepared to alter the melody, rhythms, chord progression, dynamics, etc., to best fit the lyric.
Either of you must be prepared to abandon a musical or lyrical phrase that isn't working, and recognize when "good enough" is sufficient. I've seen young songwriters endlessly polishing songs to make them "perfect", when there's just no such thing. In the course of writing, however, a fragment that's a little "off" often should be left in place to allow you to move ahead with the rest of the piece — as the point is to finish the song — then review those stubborn bits after a few days.
As for writing music for an existing lyric, that generally requires a quite skilled composer; it's considered difficult by most.
Don't worry about getting published. Songs are submitted to A/R (artist and repertoire) departments and to producers of specific artists. The record label must be contacted in advance for details of the submission process, and they will not even open uninvited material.