I am concerned about the burnt out resistor!!! The contrast control is just applying a voltage to the end of a high value resistor chain to set the voltages of the various LCD drive states. Very little current should flow, microamps only!
So figure out what you did wrong with the resistor before you do anything else.
The initialization procedure of the HD44780 controller chip is not at all complex to do. It simply must begin with setting the interface to 8 bit mode because that is the only way to set the mode to a known state. From there you can set the mode of your choice even over a 4 bit bus.
The wikipedia description of this process was a bit complex I thought, so I straightened it out a few months ago. It should be a simpler read now.
https://en.wikipedia.org/wiki/Hitachi_HD44780_LCD_controller#Mode_selectionThe problem is the controller reset does not work unless the power comes up in a specific manner which often does not happen. So you can't know if the interface is in 4 bit mode or 8 bit mode. That's ok, because the 4/8 bit mode can be set with a command sent overt a 4 bit hardware interface.
Essentially, because you don't know the starting bus mode (it can be any of three states, 8 bit, 4 bit first bus cycle or 4 bit second bus cycle) you have to send the same single bus cycle command for setting the interface to 8 bit bus mode three times. Now you can be certain the chip interface is set for an 8 bit bus. If you have 4 bit wide hardware send a single bus cycle command to set the 4 bit bus mode. Now you are in the mode that matches the hardware and you can send the full set of configuration commands in the selected bus width mode.
The timing is a bit different from what the data sheet indicates. The numbers they use are "nominal" or typical and do not take into account the variation in clock speed over temperature, voltage and process. Notice the heading in the table for timing says, "Execution Time (max) (when fcp or fOSC is 270 kHz)". 270 kHz is the typical rate at 5V using a 91 K resistor. If you use a 5V module at 3.3 volts it slows further. I think the numbers below also factor that in to be truly universal, but I don't recall for sure. Someone check my math from the graphs on page 55 of the HD44780U data sheet.
The Clear and Home commands are listed as 1.52 ms, but with tolerances should be given 2.51 ms. The remainder of commands are listed as 37 us, but should be given 61 us to execute. Hmmm... checking my math I think I did not include a voltage change. So in the worst case using a 5V module on 3.3V the timings should be further slowed a bit.