You simply did not have your weight distributed like you needed it to be for BM doubles. You may not have had any actual lead on at all, but that does not change my statement. BM doubles has more weight up at the top of the tanks compared to single tank. You need more weight down low to counterbalance it. Otherwise, what happened to you is, well, exactly what happens.
In my experience, most rec divers come into their first attempt at diving doubles with very light, neutral or nearly so, fins. For a lot of people, with no lead at all, that is a guaranteed recipe for going head down (from the weight of the isolation manifold and the extra 1st stage at the very top of your rig).
When I teach Intro to Tech (which is basically teaching doubles), the first thing I do, before ANYTHING else is get students sorted out on their weighting and the distribution of their weight so that they can hover in trim without having to scull their fins. Sometimes, that means going so far as to getting fins that are more negative. I consider that an absolute MUST. If a student has to actively work to maintain flat trim, then it will make everything after that (e.g. valve shutdown drills) that much harder. I speak from experience. When I started my own Intro to Tech/AN/DP training, my instructor did not do what I just described. He did not take the very first part of the class just to get weight and weight distribution correct and all of us in the class really struggled with our trim as we learned how to do valve shutdowns - and even simpler stuff like donating a reg, etc..
It is a shame your AN/DP instructor (apparently) did not take the time to do that first. When you have the weight distributed properly, to have motionless, flat trim, then actually diving doubles really is pretty easy.
Sidemount does have the advantage there, in my opinion. Generally, it seems to me that once you side mount cylinders so that they are properly in line with your body and properly positioned, most SM rigs, combined with the fins that most people already have, results in the right weight distribution to give you easy, flat trim. I.e. it's more common to be "okay" right out of the box with SM. When I dive sidemount, I use neutral fins - same as for single tank and my CCR. But, with BM doubles, I use negative fins. All of this being in a wetsuit. In a drysuit, my feet are floaty, no matter what, so I always use very negative fins in a drysuit.
