You can visit music conservatories, tech colegies and music instrument shops and ask to put something up on their notice boards. However, your best chance to get work is to extend your social circles into the music circles, so;
- Go to jam sessions and get to know the musicians playing there.
- Go to small local gigs and try and get to know the musicians and the managers of the bands.
With regard to not seeming desperate; in interactions with people, your job will come up naturally in conversation so you don’t need to sell anything. If it’s appropriate, you can let them know that your available for work if they (or their friends) ever need a recording studio. However, the best way to get new clients is to over deliver to the clients you do have. Be professional, focus on the detail of your work, be nice (but not pandering), deliver what the client wants. Then spend some of your own time and deliver something extra that they didn’t know they wanted. 1 x word of mouth recommendation is worth more than 1000 flyers handed out to musicians.