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That AA7EE front end is fairly straightforward.

The 1k pot is for RF gain in case you are dealing with very strong in-band signal,
the SA612 doesn't have a whole lot of dynamic range.
The tuned bandpass filter that follows the pot prevents strong out-of-band signals
from swamping the SA612, perhaps that 50kW AM broadcast station on the other side of town.
It also prevents harmonics of the desired frequency from sneaking through to the audio.

From the pot, the signal goes into an impedance matching transformer,
the output winding of the transformer is parallel tuned with 50p+47p
That first parallel tuned circuit is lightly coupled via 10p into a second very similar parallel tuned circuit,
the light coupling prevents the two of them from acting as one big tuned circuit at one quarter the resonant frequency.
The 10p also provides DC isolation, note that pin 2 of the SA612 does not want to be at DC ground.

He calls out a NE602, any genuine part marked "NE" would have to be pretty old.
These days NXP makes it, and calls it the SA612.
NXP will also sell you an SA602 if you insist, but it's exactly the same part.
There have also been "new and improved" A variants.
But I don't believe there has been a significant difference between any of these parts for several decades.

The diode tuning should be plenty stable for tuning across a few khz of 40m CW.
Add a cap in parallel with the coil till you hit the desired frequency,
adjust the 1000p cap till you have the desired 15khz range,
If not stable enough, unhook the diode, see if it remains unstable.
If not, then the neighboring caps and/or coil are the problem.
If the diode proves too unstable, make sure your "8vReg" is properly regulated.

NPO caps are very cheap and stable.
Random caps from your junkbox will give random results.

Jerry?