So let me get this straight, and work through the logic here; Its not that the engine is working too hot, or that the radiator isn't up to par, but that you have an electric fan which you switch off, so it doesn't chuck muck about the engine bay, and then get stuck, so you have neither cooling from air-flow due to motion, nor air-flow due to the fan..... and probably a rad full of silt.......
Lets think about this one, and see what we can come up with.......
Err... I think taht the starting point has to be some old fasioned 'preparation'
First, some water-proofing of the engine bay electrics so that it doesn't matter so much if water gets splashed around; coil in a tuppaware; marigold over the dizzy, WD40 on the leads, and vaseline in the plug-caps; and condoms on all the connectors.
Splash guard made from an old milk bottle for the alternator, and lots of vasceline over the spade terminals on the starter and battery terminals.
Next, if you have it, use the electric fan, preferably with a shroud, so its not so want to splash water every where, and its effect is channeled to suck air through the radiator.
and DONT switch it off unless you are actually going through a river!
The radiator itself will want cleaning of any and all crap thats in it, which will mean taking it out and cleaning it properly with a pressure washer from behind, before putting it back. worth back-flushing it inside the core while you are at it.
Next up, to stop the muck getting in it, a rad muff or shorud to go over the front of the rad panel.
This is where it gets a bit of a dilemah, becouse the simplest ting is a bin-bag opened up and taped over the rad grill, but leaving that on while splashing through puddles means no air-flow, while wading into deep water, which is usually pretty clean, you dont actually have much risk of the rad getting clogged with mud, just 'splash' around the electrics that the vasceline, condoms and marigold should take care of reasonably well.
for 'mudding' you want something that will let air through, but not much, and for that, simplest thing is some kind of fabric 'gause' like ladies stockings...... How bigs the missus' arse?
Maybe if she's a skinny bint, you might, with plenty of qualifying expolanation first, ask for one of the mother-in laws old laddered tights!
Or perhaps less embaressingly, walk into your local Marks & Sparks and ask for a pair of 'large' stockings! (muble a lot, and say that they are for your mother or something!
)
To save future embarressment, perhaps try ang p[rocure half a dozen pairs at the same time!
Anyway, stretch the material over the rad apparture in the front panel, using a bit of inginuity, and it shoudl allow air-flow, but prevent the 'worst' mud getting into and clogging the rad fins, while letting teh fan suck, should keep the air moving.