To those interested in making recordings, I highly recommend the book How to Make and Sell Your Own Recording by Diane Rapaport. To find it, click on the Support the Mudcat icon and follow the path to Amazon Books. She says

Once a song has been recorded and distributed, you have a right to record that song and to make your own arrangement of it, provided you obtain a mechanical license from the writer's publishing company. The license states that you will pay the statutory rate (or the fee negotiated) for the use of the song, based on records actually sold, and that you will make a regular accounting of all records sold and returned.

As Bill in Alabama says, the easy way is to pay the royalty (6 cents, now, I believe) at the time of manufacture (simplifies accounting).

We aren't talking big bucks here, and it is the honest thing to do. In addition, this is money that ends up in the artist (songwriter) pocket.

Bill in Alabama has good advice on finding out who published the song. If you have the song on record by someone it is usually on the label.

Roger in Baltimore