Will a new D800/600/D4 improve my work?

I shoot sports/action so when I upgraded to a D3s, yes, I noticed a considerable improvement over my then top camera, the D300. Greater high ISO capability coupled with full frame fit to my lens made a significant difference. The ability to shoot at ISO6400 and 1/500 was a huge boon.

I shoot a lot of events as well - awards, professional receptions... that sort of thing. I use supplementary flash when needed, but I'm a bit surprised that you're depending so heavily on flash for photojournalistic style "slice of life" shots given the general desire to be less intrusive. A D4/800/600 would enable you to rely less on flash because if their tremendous high ISO capability and therefore allow you to be less obvious.

If you really prefer flash in most situations, then no, I'd say the D700/300 combo you have will likely serve your needs for some time to come. You'd probably be better off investing in lens if you don't have a fixed f2.8 lens or faster.