Do I need a subwoofer? Which brand and model?


I mainly listen to classical and jazz - mostly trumpets and violins, with the occasional piano, viola or bass guitar. A couple of my friends have been suggesting to me that my ensemble lacks bass and having a subwoofer will address it. My setup includes a 2 ch. NAD C365BEE integrated amp, Wharfdale EVO 50 loudspeakers (pair), and an Oppo BDP 105 Bluray player as a CD player. The setup is in the living room, with three walls, and the fourth side open to the dining room. The living room itself would have been 18' L X 12' W X 18' H = 3888 cu. ft, but the fact that it opens up to the dining room makes the entire space more like 40' L X 18' W X 12' = 8540 cu. ft. Would I benefit from a subwoofer and which one should I get? Budget $1000 or close to that.
shugho
Post removed 
Thanks all for your continued guidance.

ZD, I believe the connections are in phase - I checked them. I rechecked the Oppo settings and changed two - one said Stereo down-mixed which I changed to Front 1-2, and the other said Crossover 80 Hz which I changed to 40. Other than that, I did not see anything worth changing. I think these made a difference in the fullness of sound, but could not be completely sure. I need a way to measure that.

Bob, Sprks, I completely agree that I need to figure out the room correction and measurement aspect before I make any changes. I am concerned that even with my current setup, which is not shabby (I think), I am not getting enough out of it. The suggestions you made will help me get there.

Others, I am open, even excited, about getting a sub and a better amp - after optimizing my current situation a little more`- and need some more guidance on the amp front. I need to determine whether I should go integrated or separate pre-power. I also need some opinions on the brand I can get without breaking my bank (3K) and would to consider something other than a NAD. I once demo-ed a MF M6i integrated which I really liked, driving Paradigm Studio 60s - thought it more musical than an Anthem integrated 225 I was comparing with. I could also consider a Krell 550i as I read good things about it. Beyond that, I am unsure what else to consider, especially when going separate.
Post removed 
Bob/All, GREAT NEWS!!!

I adjusted my speaker positions around until the zone they transmit to overlap further. Also, brought the speakers further in front and closer to each other - roughly 4 feet apart. I also re-checked the connector between the Oppo and the NAD and reset the Oppo settings back to their defaults.

Result was MAGICAL.
Fullness of sound TRIPLED.
Soundstage QUADRUPLED (just saying - not based on actual measurement).

Based upon the current situation, I don't think I need another amp. I am getting the effect I want at a quarter of the full volume.

I don't even know if a sub is required, but I will definitely give it a try to see if there is a further effect.

THANKS ALL!!!