Stick one or more linearizing inductors in series with the antenna (check their self resonance point to make sure they will function in this mode). Use 3/8" tubing to make real antennas.
I'd suggest you avoid Schmitt trigger devices. Use buffered 4000B series inverters (NOR or NAND with the other input tied hi/lo as necessary, for some reason a decent buffered hex inverter does not exist). Make the oscillators LC. Do something for static protection like a neon bulb and/or littelfuse SP721. Regulate the supply voltage(s) with inexpensive three terminal regulator(s).
That may sound like a lot, but it really doesn't take much to move a design out of the squeaker toy category.
Several things to note: ( I am presuming you are using the 4093 hysteresis - resistor [R1] from output to input, input to antenna / capacitor to ground)
1.) The hysteresis 'window' on the 4093 is quite small, which gives a low level triangle waveform on the antenna.. put a resistive divider on the input to increase the antenna voltage.
2.) Put a resistor and capacitor in series and connect this across the resistor R1
Make the resistive divider have 47k to antenna and 100k to ground - if circuit fails to operate make the 100k larger until it does oscillate. Make the TC of the series R + C about 1/4 of the period of the oscillator, and make R = 1/2 the value of R1.
The above mods will improve performance (range and linearity) a bit - but not much. Increasing the length of the antenna (as Thierry suggested) is, IMO, better than increasing its thickness - A long antenna will "see" your arm (at an angle) and the geometry helps to improve linearity.. (This is one of my quirky ideas - I believe that antenna geometry greatly affects performance of non-compensated "antennas") and think I have proved this in practice.. RC oscillators usually make unplayable theremins, but with a really long thin antenna (1.2m - or twice the arm length) one can actually play a couple of octaves quite easily) - Even better - (but more difficult, and it looks silly) is to angle the antenna towards the player.. 45 degrees gives great linearity!