Ok, updates!
I just tested two things; putting an alternate video source on the
Monitor ][, and connecting the ][+ to two TV sets. I first connected
my dead //e to the Monitor ][. Though the //e itself is not
functional, it crisply displayed all the pretty random chars that it
displays every time it boots up, no lines or anything of the sort. To
make sure, I connected my laptop to the monitor via an s-video - to
RCA adapter, and it also displayed fine. (Though windows XP looks very
odd on such a display)
I then brought the ][+ upstairs and tested it on two TVs, and got the
same result as I got on the Mointor ][, except the waves of lines were
in black and white not black and green, but that was to be expected.
To make sure that the ][+ didn't kill the TVs and that the TVs
themselves didn't have horizontal synch problems, I connected my
laptop to them as well through the same s-video cable and it worked
fine.
So I think we can now conclude that it is the ][+ with the problems,
not the displays. I am going to try pushing on the chips next, and
I'll report back as to wether or not that did anything.