Oscillator drift on presence of hand
1) may require an electrostatic screen around the inductor and (I'm going to call the right hand transistor Q2 ... you should add reference designators to all your components) Q2 base - use copper foil, or tinplate, or scraps of copper clad PCB to make a box. Soon you'll know why professional products have a lot of cans and metal dividing walls.

2) You can't take output from A ... in an ideal world there would be virtually no signal there (read up on long tailed pairs) but one of the transistors is turning on too hard, cutting the other one off.(as WhatRoughBeast notes, that's the normal and easiest way to stabilise the amplitude).

3) You can't take output from B ... this is supposed to be an infinite impedance node (ideally Q1 is a pure current source into a parallel resonant tuned circuit) and any loading on it disrupts the oscillator - not only retuning but reducing Q, destroying its stability.

You CAN put a low (50 ohms would match most RF circuitry) resistor between Q2c and 12V and take output from that. Q2 isolates this output terminal from the oscillator, providing a buffered output without disturbing its function as an emitter follower to buffer the oscillator feedback via C(argh ... the 7pF one).