I am Looking for That Elusive 3 Dimensional Room Filling Soundstage

I hate to rain on anyone's parade, but no amount of vinyl is going to fix an untreated room that is perilously close to a perfect cube.   Fortunately, these two solutions are not mutually exclusive.   But everything in its proper order. 

@dep14 offers excellent advice with respect to going slowly on room treatment.   One too many traps, or one trap poorly located, can suck the life out of a room.  With respect to the OPs goals, in my opinion the best bang for the buck is going to be addressing whatever low frequency ringing may be present and judiciously treating first reflection points.  However, there is nothing cheaper than free, so the first step is optimal placement of speakers and listening position.   Properly treating a room is an exercise in patience.  Measure, interpret, treat modestly, then repeat.  

A number of people have mentioned room treatments.  I agree.   In an untreated  room with those dimensions, you are going to have a confused mess from early reflections that really frustrate any attempts at achieving your goal.  I spent a lot of money and effort chasing better sound not realizing how important the room is.  For me, utilization of REW to discern room issues has been a game changer.  My room is still under treated, but little by little I add more treatments as I figure out where I can make the biggest impact.  I have no intention of changing any of my current equipment until the room is where it should be.  

@eag618,  I suggest you spend a good bit of time taking a look at the Realtraps website.   You will find a wealth of information on room and system set up and acoustics.  Also really useful tools.
http://realtraps.com/info.htm

GIK Acoustics has some really cost effective treatments and I think an industry leading free consultation.  
https://www.gikacoustics.com/?gclid=CjwKCAjwldHsBRAoEiwAd0JybSTTbBYpusCnPntumCDCKkolhDErBYA7oLqT7_Pa...
However, in order to make best use of that free consultation, you should fully inform yourself on the basics of room acoustics and get up to speed on the REW software for room measurement.  It is free, there is great support while you are trying to get started, and companies like GIK can use the data to identify and target your room's biggest issues. 

If you are using monitors with limited output below 40 Hz, your problem is somewhat easier than if you are dealing with full range speakers, but you still need to measure to understand your decay times etc.  

Since you are pretty good with physics, you have a huge leg up on most of us.  You might want to also pick up a text book like "Master handbook of Acoustics."