The only thing I don't like about my current ride is how far apart the pedals are. It's a kogswell frame with 135 rear spacing wearing a standard road double crankset with a BB that makes the chainline all work out. The problem is that I naturally ride the way I run, with my legs very close together, like knees brushing the top tube close together. This splays out my feet a bit.
How much narrower is a track frame that is set up with proper track cranks? Is is narrow enough that I'd be able to really feel it and get a more natural pedal stroke?
I know that track hubs are 120, that gives me 15mm right there which is a bit over a half inch. How much do track cranks move the pedals in compared to a road double?

it largely depends on the bottom bracket and the frame. some track frames need a longer bb spindle to allow a larger chainring to clear the chainstays, others don't. for example, i have a romic track frame with a suntour bb, with a very short spindle. i can run my DA road cranks with a 44T chainring on the outside on there fine, with a near perfect chainline, however, the chainring comes so close to the chainstay that i doubt i'd be able to put my sugino 75 with a 49T ring on there at all, without getting a longer spindle, which would in turn probably screw up the chainline on that bike.

however, that bike has very odd geometry, and it's not much of a problem, because i don't ride it much and consequently, it's not even built up at the moment.

long story short: yes, a track frame will allow you to have your feet closer together. the cranks won't make much difference, as the distance part of it is mostly in the bottom bracket spindle.

a few suggestions:
1) as long as you have the clearance between chainring and chainstay, and it won't jack up your chainline too bad (you may have to respace your hub) i'd say try a different bb.

2) if that won't work for you, then yeah...maybe try a new frame, or...

3) if your kogswell is steel, you could have the rear stays cold set and re-spaced at 120mm. it's only about 7.5mm for each side, and worth it if you don't want to go out and spend money on a new frame.

and yeah, you will feel it. if you've ever changed crank lengths before, like from 170 down to 165, you can definitely feel that 5mm per crank arm, so i'd imagine it would carry over into lateral adjustment as well.

I think it all depends on the frame design. Track crankarms will not clear chainstays on certain or most non-track frames. Simply using a longer b/b will give you the crankarm clearance but will throw the chainline way out and end up with a wide Q-factor which you don't want.
My track frame setup is a bit narrower compared to my road bike setup with 130mm dropout. In both bikes, the crankarms clear the stays by about 7mm...

George