Hi Dan,
I can only offer some generic hints. I own a HM 705, which is somewhat older.
At fist, I would check the power supply lines with another scope. Problems with the rectifiers and electrolytic caps are quite common. I have experienced also problems with microcracks at solderjoints at leads that interconnect boards giving strange problems, that looked like temperature dependent.
Next, I suggest using a plasic toothbrush to brush around to locate the area of the problem.
Next step is using freezing spray on single components.
Use of an isolation transformer should be mandatory.
Good luck,
Chris