There seems to be some sort of urban myth that road bikes are made of fairy wings and tissue paper, they are plenty robust enough for a bit of a punishing road surface. Put some mudguards on (my 700x23 frame has absolutely minimal clearance & no lugs but still takes crud roadracers happily enough) & mud is no longer a problem, soften the tyres a few psi and you increase the comfort over the bumps. But CX are corking too, unless you're really thrashing along, the speed differential isn't massive & the geometry is a little easier but still gets you down and out of the wind when you need it to
CX's usually have a higher BB for better ground clearance too but that doesn't necessarily equate to needed for bad country roads, more bouncing over tree roots and rocks on off road courses
I ride different bike types on bad quality roads & country ones frequented by tractors and horses so usually not clean at all & TBH have no more worries with my roadie than my hybrid or MTB. I occasionally take the roadie through my country park commute on decent quality mud & gravel paths too & its still alive.
You adapt your pace according to where you are, if the roads are that bad, you're not going to be pelting along and banging through potholes on any bike, however robust, without consequences.
Slick tyres vs CX knobblies. are you talking inches deep mud through which you never actually touch tarmac - in which case bite the bullet and buy a CX - you'll need the frame clearance for suitable tyres
Or the usual field wash off and tractor tyre deposits that you still make contact with proper road through - in which case take your pick, slicks will cope & common sense is the order of the day, don't be making sudden direction changes mid gloop and you're good - again though, overcook it on any tyre/bike type and you're on your backside in the road.
If you do go for a CX, have a look at the Edinburgh Bikes 'Cross' range for competitively priced, remarkably well specced for the money and very useable CX bikes - I had a 2010 base model and loved it - someone else took a shine to it when my son parked it up one day too though, never got it back.
