Water – Which to replace – the water heater or the 3-way switched plug adapter or something else

It sure looks like you are on the right track with you assessment. Most electric water heaters have two thermostats, one that controls the main tank element, and one that controls the quick recovery element. There are several very helpful test procedures available online or in the This Old House website. The basics of the test are very simple if you have knowledge of using a volt/ohm meter.(VOM) Be absolutely sure the breaker and power to the heater is off before attempting any OHM measurements on the elements. You will check for proper operation of the therostats. This is usually done with the power on (220VAC) , which can be dangerous if one is not competent in testing live circuits with a volt meter. Basically, you will first verify the 220 volt input at each thermostat. As you draw down the hot water in the tank,replacing it with cold water, the output relays(just a double pole switch for testing purposes) in the thermostats will close, putting 220VAC on the output terminals to the elements. You can also simply turn up the temp on the thermostats and check the outputs as you are now asking for hotter water. Assuming you verify proper operation of the thermostats, the next step will be checking the resistance of the elements with the ohm meter. BE SURE POWER IS OFF. Disconnect the output wires from the thermostats and measure the ohms of each element. Knowing the factory spec is very helpful, however a very high reading or a "0" reading will indicate a bad element. I will try to find a good link and post it in the comments section for you, but a good procedure is easy to find as stated above. http://homerepair.about.com/od/plumbingrepair/ss/element_replace.htm

Without natural gas or propane, your only source of heat is electricity. Electric tankless water heaters would probably require a new circuit to be run. Here's a table of typical power and wire requirements:

http://www.rheem.com/documents/rte-13-tankless-electric-spec-sheet

The smallest unit they recommend for a shower is the RTE13 which takes 13 KW or 54A@240V. This requires #6 wire. If your incoming temp is 40F and desired out temp is 110F, you need 70F of temp rise and the RTE13 will provide 1.27 GPM.

Many people think they want tankless because the mfg tell them tankless is more efficient. There is some truth to this for gas fired water heaters but in the case of electric water heaters, the difference in efficiency is very small.