Most complex machines use timing belts to keep different components working together synchronously. When timing belts fail, it's usually a bad day (eg, the car engine seizes up). Something as important as an airship will probably have fail-safes in place, as any one timing belt would create a single point of failure, and it's not like you can just "pull over" (in a car), or pause the engine (in a boat) to fix it. The airship demands continuous action.
So what if, instead of something failing it gets off - but only slightly. These types of timing problems can be very tricky to detect. The larger, more complicated the system, the more difficult it may be to detect.
But it gets worse...
You mention automatons working in the engine room. What if they're all just slightly off? This could start by causing engine strain, a blown gasket here or there, and so on. The more off-sync the automatons are, the more catastrophic the problem. Ironically, this means the timing failure becomes it's own type of clock - if the timing issue is increasing by one millisecond every hour, and catastrophe occurs at ten milliseconds off, then in ten hours we're all screwed.
Solutions for timing issues are never easy, but the best way mechanics do this is usually "stop the engine, replace the belt, try again". Can't do that on an airship! Furthermore, even if one component can be stopped the idea of "stop the component, replace the belt, try again" may not work - the automatons are still out of sync! So the real question becomes "how do we get everything back in sync - and without stopping the automatons?"
Fixing the problem
Possible solutions include manually adjusting one piece at a time, forcing a failure on a component (forcing the backup to take-over), replacing an automaton with a human temporarily (assuming the human has good timing, and there are enough humans!), and so on. This helps with your central conflict, though, as you might not be able to try multiple solutions at once, so one of the two characters will have to give in.