I have a James and Elizabeth Alexander in my ancestry and the dates you mentioned, 1794 and 1803 are congruent with the information I have.The only thing I do not have is information of an eldest son named George b. 1754.The earliest record of a child that I have is that of their daughter Mary b. about 1760.
Another thing that greatly interests me is the mention of a Robert and William Alexander.In reading through some of the research my mother had done on this line years ago, those names appear.Unfortunately, I don't have details off the top of my head at this moment, but would be happy to find the information and relay it to you if you would be interested.
I am stuck here at the James and Elizabeth Alexander point.I have no further information.I would appreciate any information you may have if we share this line.
Oh yes,there is also a Micheal Ehart in my line.He is the grandfather (or could be father) of James and Elizabeth's daughter-in-law FrancesEHART(she married James Alexander Jr.)
Please feel free to contact me via email at firstname.lastname@example.org.I look forward to any replies!