The idea of getting your speakers into the room should address one thing in a big fashion. The rear-ward wave will hit the wall. The distance that wave travels tells you something. If your speaker is 4' away from the front wall, you will get a null at the 1/4 wavelength. That is, 1130 ft/s divided by 4' multiplied by 1/4. This yields roughly 70hz. You will get "issues" in the FR at every multiple of that 70hz in addition to the "issues" associated with the 3 dimensions of the room. If you have your speakers 3' away from the front wall, the problem is roughly 94hz. As you go closer to the front wall, the problem frequency increases into the vocal band. In my opinion, Rock music suffers greatly when speakers are close to the front wall. Having a null in the 100hz range hurts the grunge of the electric guitar and makes the rock music lifeless and boring. In small rooms you will trade off bass drum definition and rock guitar grunge. This is where a subwoofer can help, even though you have full range speakers. Get the guitar grunge back, use the subwoofer placed appropriately to get the bass drum back.
The reason why its necessary to make changes slowly and 1 dimension at a time is the following:
1. 3-D room = 3 room modes. Problems at these 3 frequencies and their multiples
2. Floor bounce. If a driver is playing in the range that includes the frequency that correspondes to the distance between it and the floor, there will be an additional 1 problem frequency and at its multiples. Some manufacturers dial this out through the crossover network, some dont.
3. Rear wave. Described above, 1 extra problem frequency and at its multiples
4. Side wave. Similar to above, 1 extra problem frequency and at its multiples
This is a summary of the major problems. As you can see there are 6 frequencies in the 47hz (length mode) through 140hz range (side wall/ rearward wave). If any of these frequencies overlap, they make the problem at that frequency much worse. This holds for any of the multiples as well. This also does not take into consideration tangential modes or oblique modes of the room.
So, how can one get a great sounding system with so many problems? One concept is a large room. Larger rooms have lower modes. The lower the mode, the more multiples you can fit within the spectrum. By having every frequency being a problem frequency in the absolute sense, this also means that the problems at each frequency are relatively small. In order to get close to this absolute, you have to ensure all room dimensions are dissimilar, the placement of the listening position is dissimilar to any of the room dimensions, same with speakers. Similarity is "defined" as divisible by 2 or 4, etc.
So, the take home message is if you have a smaller room, you have to ensure that your speaker placement and listening position placement does not overlap any of the native problem frequencies that your room already has. By spreading out problem frequencies, the problems become easier to deal with, especially with acoustic traps.
Its gonna be a lot of work, but this hobby is all about that
One final note, in my experience I need to be at least 4'-5' away from the speakers. You might have to sacrifice the 38% rule towards 33% to get this distance as you pull your speakers forward. I would pull the traps out of the room first though. Otherwise you have to rearrange the first reflection traps for every move you make. Also, without the traps, every move you make will cause drastic audible and noteable changes that you can write down.
While you might change out your speakers, this is still a valueable excersize as what you learn now will carry to any of the new speakers. If the new speaker has compensation networks for boundaries, your life is made even easier. However, why use those networks if you can optimize placement first? You would only know those optimized placements if you had done the work previously in that room

This is tedious, thats why acousticians get paid a lot of money to "make it right". Usually, if you are saving money (by not hiring an acoustician) you are going to make up for that in time costs