I am sorry for being late to the party - the house was burning here.
In my 34401A, the experience was that a missing Gowanda inductor in the input stage caused very, very similar behavior. I replaced the U201 opamp at risk at first, which turned out well (surprisingly, given my uncalm hand) but did not solve the problem.
The 3xxxx series measure their resistance via the voltage. So you need working DC voltage meas, before looking at fuck like current source. Oh, and sth evil: current path can be OK while voltage path is fucked.
P.S. If you need more on that urgently, pls PM me to remind me to come back here. The next week will also be hell.