Just to followup on my own posting....
The problem is a little better now and the main change that seemed to have an affect was to increase the HTTP KeepAlive Timeout from a value of 3 to a value 60. This seems counter-intuitive to the Moodle Performance docs, so I thought I'd just post it. Once we made that tweak, the ICMP messages being generated from the server dropped dramatically. There are still slow responses being reported by the students - but not as many.
One thing different about our site is that the whole thing runs in HTTPS. We use mod-rewrite to force all Moodle work to be done in HTTPS (this was a requirement so that the students could exchange info securely).
I still welcome any thoughts. Thanks.