Counter glitching? Instability? Can't get a trigger lock on the scope, what might that be?
Indeed, as the commenters asked, the best explanation for counters glitching after everything else on them is connected right, is that the outputs are connected to other components that have something wrong with them. Outputs connected to outputs. And it turns out that meticulously tracing it through disconnecting all the wires and connecting them, and even using an analogue Ampere-meter I found that there was current coming out of an input pin of one of the following 74LS652 bus transceivers.