Is my 2-gang switch box over-full with 5 cables?
Your wires and devices need 18x2.25 = 40.5 cubic inches. 

What you have there is a 4x4 square metal box.  If it's 1-1/2" deep it'll be 21 cubic inches.  Otherwise I'd expect to see a cubic inch number stamped onto it. There's also a 2-gang mud ring of, well, it sure ain't 19.5 cubic inches!  

Note that the mud ring does not come all the way out to the wall surface.  It is quite shy of that.  This is improper installation, so you need to replace the mud ring anyway.  So replace the mud ring with one of the correct height, even if it doesn't have enough cubic inches.  We have one more trick. 

Lastly, you can obtain a surface conduit starter box e.g. Legrand Wiremold.  That will provide some additional cubic inches (depending on its height). 

If you get the statutorially needed cubic inches, you should have enough room for the dimmers.