Mysterious ground wire
The cure for this is a surface conduit starter box.   This box will sit on top of the existing box about 1-1/4" proud of the wall.  It will provide both the statutory cubic inches you need for all those wires and yokes, and the practical cubic inches for your dimmers.  

Normally it's intended that you then use its sides as a launch point for surface conduit, but if you don't do that, I won't tell :) 

Now, I can't guess what's going on with all those black and white wires.  I surely hope you can.  Normally I expect to see some of them be wire-nutted to each other, and this positional data is very important. Usually the only evidence that tells you which wire does what is how they are connected in the box.  I gather two of the blacks are switched hots, and the other two are supply and onward, which need to be connected together obviously.

With those ground wires, I would tuck 3 of them back, and send the fourth into a wire-nut to meet the pigtails from each switch.  

Now as far as cubic inches statutorially, you need to count the wires and the "wires".  

I see 8 wires in there - 4 pairs of black-white.  I also see a bunch of grounds - all grounds together count as 1 wire.   If any of the switches have pigtails, pigtails do not count.  Each switch (or to be more precise, yoke) counts as 2 "wires".   So I see a total of 8+1+2+2=13 "wires" in the box.  

#14 wires require 2.0 cubic inches each, for 26 cubic inches.  #12 uses 2.25 cubic inches per wire, or 29.25 cubic inches.  The 24 c.i. plus a starter conduit cover should get the job done.  

If you want flush mount, you must replace the junction box in the wall. What you want is a 4-11/16" square box with a 2-gang mud ring of appropriate depth.  This will be bigger than your normal box, but will undoubtedly fit the two large dimmers.  You can try getting it done with a 4x4" square box with a 2-gang mud ring, and this will fit in the footprint of the existing box... but it could be a tight fit.