Slow Running/Backed Up/Soapy Laundry

I recently relocated the wall behind my washer and dryer, along with the plumbing and drain. I see a few people having the problem I'm having now with soap backing up enough that it drips. When I replaced the 1 1/2" pipe I replaced the trap as well. But where there were two traps attached to each other, I replaced it with one. And while I had soap back up in the past, it was always minor. I'm fairly certain I need a trap, since this goes directly to the main line. I wonder though if I wouldn't benefit from putting a bigger drop in before I install the trap. I haven't enclosed the wall yet, and still have the freedom to fix the problem, but wondered what the best fix would be. And looking back on the two traps that were installed before, I wonder if the double bend doesn't prevent soap backup as well? Anyone know the recommended approach? And since it's all new pipe, I'm inclined to believe that others having the same problem won't benefit from snaking. I think it's more related to design.

Washers today put out a huge amount of water very quickly, so they require a 2" drain. And, yes it requires a trap and vent. If I read your post correctly, you had an illegal trap before if the two traps were connected together. I think that would have made an S trap. If you can replace the 1-1/2" pipe and trap with 2" into the main drain and vent it, you should be fine.