... doesn't really help me if I want to play with a drummer. If I turn the amp up to where it can be heard (call that 3) then it is very overdriven. ...
This is where you have to figure out whether the amp is distorting because the EL84s are distorting (and the EF86 is clean), or because the EF86 is distorting (and the EL84s are clean).
Fastest way to do that is measure the d.c. volts across the EL84 cathode resistor: that's your bias voltage. Now attach your meter to the EL84 control grid, set for a "peak a.c." measurement. Play until distortion just-starts, then back off a hair on volume. You might need to reset the meter's peak-hold function to capture the peak signal driving the EL84s just-before distortion starts.
Compare the measured peak to the EL84 bias voltage. If the peak voltage is about the same as the EL84 bias, you're hearing the EL84s distort; you'll need a bigger output section for louder-clean. If the peak voltage is well below the EL84 bias, you're hearing the EF86 distort (or much less likely, the phase inverter distort); adjust the screen voltage to reduce distortion.
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Which way, or how do I adjust the screen voltage?"
Put a pot somewhere: might be as a variable resistor in place of the resistor from B+ to screen. Or it might be that you keep the screen resistor as-is, and add a variable resistor from screen to ground/cathode (somewhat like the vintage AC10).
You're adjusting by-ear to your-taste, so no one can really help you out with what to do. Just gotta do it, make a choice aligned to your preference.