Well pump ends up overwhelmed. How can I solve this?
Help your wife to become at least something of a country person by letting her fix it while you (patiently and nicely) explain what needs to be done. Either of you should write down notes and/or take video or pictures, it being the modern era, to help with doing it unassisted. Then practice a few times without your help to build confidence and figure out what else might need to be written down or videoed.

With the pressure off the water side of the system, the tank (assuming a modern bladder-type tank) should be a few pounds below the "cut-in" or start pressure of the pump. So, if the 26 PSI (I evidently misread that - 28 might want to be adjusted to 27) measurement was with the water side drained/no pressure, that's roughly correct. The point of this difference is to allow water to keep flowing to the house as the pump is being started. 

The exact nature of your problem may affect the right solution to solving it. 

A flow restrictor as Jack suggests is one approach, to limit the water taken out of the tank to a rate the pump or well can keep up with. 

If the problem is the speed the pump can provide water at, pumps can be replaced with pumps of greater capacity. You can also provide a larger, or additional pressure tank to increase the stored capacity for temporary loads. 

If the problem is the speed the well can provide water at, that's a MUCH more expensive problem to solve, and far more unknown. Drilling deeper can cause more problems than it solves, depending on the specific (and somewhat unknown) details of the hydro-geology at that exact point, for instance.