Adam, welcome and a very good question.
In 20 years of driving these glorious cars I have very rarely put the side screens on or roof up, I don't set off in the rain but if I get caught just carry on.
I once got caught in an absolute deluge around 100mls from home on an afternoon jaunt and with no options carried on and got fairly well soaked and the only way to dissipate the 2" of water in my footwell was to lift the carpets out and drill a couple of 1/2" holes in the marine ply (plugging after) with removable plugs.
I always treat the leather with a good coating of dubbin and mop the dash as you go, wearing glasses is not an asset with rain but well worth the fun of it all.
There are as many methods of using these cars as there are options on a Morgan, chose yours and drive
