These power supplies are easily repairable; just a matter of tracking down the problem.
Backgrounder (just in case): The current knob sets the maximum allowable current that the load can draw. However, if the load requests more than the set value, the power supply will lower the voltage to meet that set value. For example, if the voltage and current settings are 5V and 0.10A, respectively, and the load is a 100R resistor, then the meters will show 5V and 0.05A. However, if the resistor is replaced with 33R, then the meters should show 3.3V and 0.100A. With low value resistors or shorted output, this will result in a voltage of 0V (or close too it). Same for full CCW current setting.
Given the above, 10K across 12 volts should result in 0.0012A (Ohms Law) which is too low a value to be displayed by the ammeter display; only good to 10mA increments. Check with your DMM again. You may have misread the scale.
As for your testpoint readings, be aware when reading the schematic that it is best to think of this power supply as being a negative supply. What I mean... when probing with your DMM consider the positive output post to be 0V and the negative output post to be variable from 0V to -15V. Why? Because, the bipolar power (U2, U6) supplied to the op amps is referenced to the positive output post (see the [+S] symbol, aka. TP6). Other an TP2:TP4, all your other readings depend on the voltage knob. Instead, you want to know: TP1:TP6, TP8:TP6, TP5:TP6, TP7:TP6. BTW: TP2:TP4 seems reasonable for an unloaded+rectified+filtered secondary (at peak) of 28V since the supply is spec'd to provide 15V@2A. See attached.
The 3-legged things attached to the heatsinks are U7, U6, U2 voltage regulators. Not sure about the "yellowing resistor blob" (C10 maybe?). What's the text label beside the component say?
As for the "counting down affect"... I would start with checking TP5:TP6 and TP7:TP6 are stable when the CC Set is pressed vs. not pressed. Then work your way out from there. Like checking if the counting happens with a load attached (with full CW on current knob), and comparing with your multimeter (ammeter mode) in series. It could be just the meter PCB or extend to the CC Error Amp and beyond.
Also, I just remembered, my E3611A needs 5-10 minutes after cold start for the internal reference (U3) to settle down (drifts downward a few 10s of mA with the CC Set pressed). I've been meaning to change that out for a new TL431 (+ resistors) to replace the LM336 (U3). Maybe you have an especially bad one. Does the counting down level-off after, say, 15 minutes? The place to check is TP5:TP6.
HP E3610A Service Manual