Hi there,
Pleased to hear your story and for you to be so honest about what happened which is commendable. There are a few lessons here I think to take away from this, no doubt everyone will have there own idea, but here is my take.
1. The alternator light is a land as soon as practical item, I.e. divert to a base where maintenance may be available, but also extended flight is not recommended as assuming its not working you have 30-40 minutes battery power or so.
So I personally would have done exactly that i.e. diverted, deciding yourself that it was OK even thought the light was on was clearly a mistake I think.
You didn't say how long you flew like this but I am guessing it was for an extended time ?
2. When you noticed the tachs were high although no change in sound, this is the part that can be v.dangerous i.e. overreacting to benign indications, and in the situation you describe if you form the judgment that both tachs are misbehaving you should in accordance with POH allow the governor to maintain RPM and again land as soon as practical, but I would agree that if you think the RPM has increased from a governor issue that again in accordance with POH that switching governor off that cautiously reducing throttle taking note of the sound and low RPM horn is not unreasonable but in this situation with no RPM information and not knowing what is going on you should be landing immediately.
So think your decision to land in the desert at that point was good, but probably overdue.
3.
I make the decision to continue flying for 5 minutes to the airport. Using the sound as my new tachometers, I lift off and go to the next airport. Nervous about stalling the rotor, I keep it on the high side.
I think this was a v.poor decision, there is a limitation in the POH that you must have a functioning alternator and to delibarately takeoff without RPM information and other systems is dangerous in my opinion.
You didn't at this point know if you had oversped it or no so it needed to be looked at by an engineer before it flew again, in my experience of over speeds many have been by private pilots gripping the throttle too hard and having the ERPM/RRPM off the top of the clock and causing damage to the aircraft.
Overall I think you need to follow the emergency procedures as set out in section 3 of the POH, and the limitations in section 2 more closely.
Regards CF