Actually, neither 'speak *bad' nor 'say *bad' is good English. That's not oversimplistic, it's just standard grammar.
Idioms like 'speak ill' and 'do good' are exceptions to the typical English pattern, because 'good' and 'ill' are longstanding nouns in certain contexts, even though in many other cases nouns with the same form as adjectives simply don't exist. To express concepts like 'Schlimmes,' 'Interessantes,' 'Politisches,' we usually have to use the word 'things': bad things, interesting things, political things. (Or a translator may choose other abstract nouns as appropriate: bad consequences, interesting ideas, political issues.)
'Bad' may be something of an overlapping case, because there are contexts such as 'The good and the bad in life' where the adjective can serve as a noun meaning 'bad things.'
But with verbs like 'speak' and 'say,' 'speak *bad' and 'say *bad' are simply not idiomatic -- indeed, in the mouths of people like Trump and Sanders, those errors sound pig-ignorant. Not 'North American informal,' pace the descriptivist BE dictionaries, who seldom have much feel for correct vs. incorrect AE usage anyway; just wrong.
In case of confusion, it may be easier to see the pattern with another adjective like 'negative' or 'critical.' It's still obvious (to most educated / thoughtful speakers) that it must be 'speak negatively' or 'say negative things,' 'speak critically' or 'say critical things,' etc. (Unfortunately it's no longer obvious to everyone that it also must be 'take something seriously,' because of a stupid pop song, but that battle still should be fought as well.)
Similarly, it must also be 'speak badly' or 'say bad things.' And since 'speak badly' can also mean 'speak poorly / ineptly,' we can still prefer the unambiguous longstanding idiom 'speak ill (of someone).'
So there is no need for learners to be confused, even if some dictionaries are unhelpful at distinguishing correct from incorrect usage. And there is also no need for anyone to take Trump, Sanders, or anyone else who speaks like that as a model.