From John, WB4LNM, "One of the Hams from Heath":

Here is a (unsolicited!) guess that you might want to check out:  None of
the HW-16's three oscillator tubes, V5B / V3B / V2B have any voltage
regulation on the plates at all.  Any movement of that plate voltage up and
down with strong signals is likely to make one of the oscillators move in
frequency, especially the L/C VFO, V3B.

I guess I'd put a voltmeter, preferably a VTVM or FET voltmeter, at pin 1
of V3 and take a look at the voltage while you are receiving a signal that
seems to cause the problem.   I suppose there are a couple of things
possible if you have already checked the plate voltages on the three
oscillator tubes and found them nominally okay and the three carbon comp
plate resistors not changed in value (they do frequently, but you most
likely know that already).

The HW-16's audio output tube, V6A/B, draws a lot of current, and probably
pulls more on strong signals than weak.  Its supply voltage is ahead of the
oscillator in the power supply, thus any increase in its current used is
going to lower V6A/B's plate voltage, and the oscillators plate voltages,
too, including the VFO.

Truthfully, I'd never noticed until your comment that this VFO plate
voltage was an unregulated voltage!  When I do get mine back out and
operating, I may put in a Zener diode regulator around the R33 / V3B pin 1
area.  5 Watt, 100 volt, Zener diodes are $0.57 each at DigiKey, and could
keep the VFO plate voltage stabilized much better than it is supposed to be
only adding about 15-20 mA of Zener current (1.5 watts).  I see the plate
voltage jumps 10 volts between transmit and receive and that isn't good!
 R33's value would need to be changed for this.

The same voltage stabilization could be done for the HFO and BFO, but since
those are crystal controlled, they are less likely to be the problem.

The VFO V3B appears to be drawing nominally 4.5 mA, so R33 is dissipating
0.95W nominally for a 1 Watt rated resistor.  Good, but not a lot of
headroom, and I bet it has cycled quite hot and cooled many times during
this rigs life.  47K at 2W would have probably been a better choice, and
after years of use, I'd be surprised if R33 hasn't changed value (upwards)
significantly, causing the plate voltage on the VFO tube to be much more
sensitive to power supply voltage changes.