Resistor 1R is used to sense and limit the current flowing through the MOSFET. Interrupting this current (inductor current) causes a large voltage spike when the MOSFET is switched off. The higher the current, the higher the voltage spike. The idea is to have the voltage due to the current flowing through the inductor (measured across the resistor) iL(t)peak*R) = Vramp so that (iL(t)peak*R) - Vramp) = 0 . Increasing R from 1R to 3R results in an extra voltage of (iL(t)peak*3*R) - Vramp) = ((iL(t)peak*3*R) = (iL(t)peak*R))) = (iL(t)peak*2*R) = (2*V(1R)) during every MOSFET on-time. The controller would normally limit the peak of this current-representing voltage to Vramp. This means that the V(1R) = Vramp is attained sooner, about on-third the designed MOSFET on-time (duty cycle), so obviously, you have changed the duty cycle of the converter and as such the current flowing through your MOSFET after replacing the 1R-resistor with 3R is one-third the value that it was designed for and ,hence, the voltage spike is also reduced. Consequently, the current transferred to the secondary is reduced by the same factor (and, of course, the turns ratio). But the current you are measuring at the output is an average value of this peak current. It is very likely that your MOSFET isn't failing any more after this replacement and that is because the voltage spike is now probably within maximum Vds rating of the MOSFET.
The replacement, however, tend to fold back the output voltage for the same load reduce but the controller doesn't want to see that. As a matter of fact, duty cycle is a function of the input and output voltages of the converter and perturbations in these voltages are what should cause the controller to change the duty cycle. Now that the duty cycle is altered not by these voltages, the controller is having a nightmare. And moreso that this alteration may have changed the mode of operation of the converter all contribute in causing the controller to suffer and so heating up.
My recommendation:
1) Check the snubber components. You may find one that had gone faulty.
2) If you eventually find one, reinstate the 1R-resistor.
3) If none is faulty, then, I don't know; [I scratch my head] the transformer? The...?[I scratch my head again]. Maybe, be patient for more help to come...maybe, throw away...maybe give more information.
I think Vramp for UC3842 is 1V. I might be wrong, I haven't checked the datasheet though. If it is 1V, then that's 1A peak-current through 1R (and you've reduced it to 0.3333A+ with 3R.
What are the charger nameplate info?