If you have music written for your words know this: The great misconception among aspiring lyric wrters, driven by ads in the back of magazines, is that song publishers are searching for lyrics. They're not, at least if they're reputable. Those ads target people who are uninformed and the "publishers" running them operate a shady business outside the normal music industry. The melodies they use are not custom, they're stock melodies that are re-used over and over again on new customers. The resulting songs are worthless.
Legitimate, reputable song publishers review favorite things lyrics complete song demos- both music and lyrics- and expect a somewhat professional presentation: no hype, song lyrics neatly typed, and most of all, a demo that is up to industry standards.
The catch-22 for most lyric writers is that it's very difficult to find professional melody writers in most local areas, yet they must have a custom melody to even get a hearing by an industry pro. And most pro melody writers won't work with anyone who hasn't yet had demonstrable success.
Why wouldn't a professional local musician want to write a melody for your lyric? Because they probably play music live or do studio work full time. Any personal time they have is spent on their own projects. To spare feelings they may say they'll work on it, but they probably won't.
A few suggestions:
1. Try humming your own melody. Most demo services, mine included, can work with an mp3 file of nothing but you singing a capella, letting your melody suggest the chord arrangement.
2. There are online songwriting forums where people sometimes volunteer to write melodies to lyrics. It's difficult to imagine a talented melody writer/musician spending time on a new writer's lyric when they could work with peers or spend time on their own lyrics but it happens.
3. Some demo services, such as my own, will write melodies as either a work for hire (fee based) or with no upfront charge (an agreement to split royalties is typically required). If you go this route be certain the melody will be custom written for your lyric and definitely ask for samples of their previous collaborations. If they are unwilling to furnish samples it's a sign they aren't very proud of their work.
Hopefully one of these methods will get your lyric turned into a complete song and demo and you can move on to the exciting marketing phase: marketing your song to music industry producers, song publishers... maybe even directly to your favorite artist!