Question about running new drain pipes in Laundry room
If it were me, I'd install a new sanitary tee in the main drain, just above the existing sanitary tee.  I'd have the standpipe for the washer come down into a P-trap, and connect the other end to the new tee. 

However, you'll have to make sure the standpipe is at least 18" above the trap weir. If that height cannot be maintained, you'll have to move some things around.

I'm also not sure if a proper P-trap will fit in the stud bay, so you'll have to measure to verify.
 The minimum length of a trap arm is two pipe diameters. So if you're using 2" pipe, that's 4" from trap weir to the sanitary tee.

Then I'd have the trap from the washtub, connect to the existing sanitary tee.

Basically, something like this...