You can use a JFET as a variable resistor for signals less than about 1/2 a Volt. The trouble is that their parameters are so loosely defined that you'd end up spending all day measuring individual devices and trying to find three that match.
Another approach would be to use LDRs (light dependent resistors). They're not very well matched either, but you could get reasonable tracking by adding trimpots to adjust the low and high resistance of each LDR. Yech.
Plan C is to use digital potentiometer chips. No problem with that except that the resistance isn't continuously variable, it increments in discrete steps.
Or you could just use a 3-gang potentiometer. You can't buy those, but you can buy seperate wafers, shafts etc and make your own.
----------------------------------------------
However all of those solutions involve too much hassle, expense and complication for my liking, which suggests that any circuit requiring three pots is a bad idea.
Another slight problem with your circuit is that the three variable resistors have to be well matched. If they're not, then the gain of the circuit changes. Presumably you were going to make Rf or Rg adjustable so you can set the gain to a value that gives stable oscillation with minimum distortion. However you don't want to have to re-adjust the gain every time you change frequency because the three pots don't track well.
----------------------------------------------
I suggest using a circuit like the one below instead. It only needs two potentiometers (VR2 and VR3) to set the frequency so you can use a normal stereo potentiometer.
As an added bonus, the outputs of the four opamps are in quadrature, with equal amplitude, which may come in handy.
Because it's based on all-pass phase shift filters, any mismatch in the capacitor or potentiometer values just causes a change in the relative phase of the outputs, rather than affecting the gain.
VR1 is a trimmer to adjust the gain. In the pic below, some of the waveforms are clipped because I set the gain too high in the simulation. It could be a lot better but if you want the lowest possible distortion VR1 should be replaced with some sort of AGC control.
Also, i would like the resistor values to be from 6ohms to 10k.
That's a bit of a tall order. The normal approach is to use the potentiometers to adjust the frequency over a range of about 10 to 1, and use a switch to select different pairs of capacitors for different decade ranges.