JC, the problem is with a faulty component in your Light Control Module. I had this problem and have seen numerous posts from others with same problem.
There are no relays on the E39's for switching the lights. They use power transistors located in the LCM which replace the need for relays and do the same job. You can go directly to your dealer and pay them upwards of €500
to replace the module, or if you or you know someone who is handy with a soldering iron, you can replace the component, like I did. It took about 2hrs and the component cost €5.
If you can repair it yourself, get back to me and I'll ty to give you step by step instructions on what to do. The component in question is manufactured by Infineon and is a BTS425L1. I found when I went to get one, is that they were obsolete and have been replaced by BTS426L1. From what I remember, it had 7 pins and there was 14 different MOSFET transistors in the module for all the lights. It was easy enough to trace it but a bit tricky to remove it with all the heatsink mounted around it, but not impossible by any means.