Heh -- if you have a hammer, then every problem looks like a nail.
Unless there's evidence of the problem being bad capacitors, don't fixate on it. You have an ESR meter. They are *very* good at finding "bad capacitor" problems. The other tool is your eyes -- are any capacitors suffering from bulging tops, leakage, or have been pushed over by pressure from within? If the answer to these are NO, look for something else.
Generally the classic "bad cap" problem is limited to applications where a low ESR is a requirement. These are places where there is a high ripple current and you find 105 deg rated caps (although it's not universal -- especially with poor design).
I've *rarely* seen them outside of circuits handling power at moderate frequency (switchmode regulators are the classic). That's not to say they don't fail elsewhere,
So having covered that, *AND* the fact that you've checked them with an ESR meter means that we can probably eliminate that as a cause for the moment.
If you have a mosfet and a resistor that seem to be overheating and being destroyed, then you need to look at both the load and the gate drive. The fact that the resistor (is it in series with the mosfet's drain or source?) is affected seems to me to point to a problem in the load rather than the drive to the gate.
If the gate wasn't being driven sufficiently hard to turn the mosfet on or off totally, then the mosfet could overheat and die. If it ends up as a short circuit, then I would not imagine the resistor would be affected, unless it is not rated for a continuous load (and the circuit diagram should reveal that).
If the load is short circuited (or the supply rail is high, or any manner of other conditions) then the mosfet could overheat, and then (or simultaneously) so might the resistor.
All of this is speculation until we see the circuit though. But it's an indication of how you might be thinking if confronted with this fault.
edit:
It would be useful to know what failure mode both the mosfet and the resistor exhibited. Was the mosfet shorted from drain to source? Was the resistor open circuit? (I presume we'll find the type of mosfet and the value/rating of the resistor from the circuit diagram (otherwise those too please).