You've got it. But, some thoughts:
A full frame DSLR will give you maybe 2 to 3 stops over micro four thirds - not a path you are likely to take if new - but see comments on D700 below.
It would be useful if you cited actual apertures and focal lengths as these are probably specific to your case.
Presumably "standard zoom" is ~+ constant f/2.8.
Presumably faster prime is ~= f/1.8 or 1.7 or just maybe 1.4 or 1.2
At the low light limit optical stabilisation makes an immense difference. Even 2 stops can take you from 1/15s to 1/60th - with the former liable to blur almost anything at typical focal lengths and the latter often being usable.
As well as the game changing flash you might consider a 'video light' / LED bank / ... extra illumination source which does not provide the immense gains of xenon flash but which may eg give you 2 to 3 stops with relative ease and OK cost, and you can tailor light colour temperature to suit yourself.
You say "slow zoom" so probably say f/3.5 - 5/5.6 range?
Even an f/1.8prime gives you 2 stops over a f/3.5 nd is worth playing with to see if the gains persuade you to reevaluate "your kind of photography".
The 2 stops from a f/1.8 prime is a very useful gain. The downside is the immensely small depth of field in most cases at f/1.8. While this is often why such a lens is chosen, you can get too much of a good thing :-). At medium to shortish distance a 50mm f/1.8 on and APSC body (crop factor of 1.5:1) makes it hard to get all of a face in focus if other than straight on.
Get a better body: Consider using your existing equipment as now and adding a separate camera and well suited lens that meets your low light needs. Compare the price of doing this with that for buying better lenses for your existing system - the difference my be low. The Nikon D700 can be bought at very good prices if in well used condition. Value for money needs to be assessed on a case by case basis.
I use a Sony A77 for as much as possible due to its superb abilities as a picture making system and in-body stabilisation. When it comes to taking low light photos a Nikon D700 utterly trounces the A77. In situations where ultra-low light 'may happen' but is not the overwhelming need (stage show, dancers, ...) I tend to carry the A77 plus a D700 with a f/1.8 50 mm lens (the cheapest lens Nikon sell). The D700 lens is unstabilised - while stabilisation would be mice, I usually find that ISO 3200 and f/1.8 produce results that are acceptable in many cases. The D700 can be used at ISO 6400 but is getting decidedly noisy. FWIW the D700 has a per pixel noise level better than any more recent DSLR except the D3S.