My hunch is that 'straightly' should probably be marked [rare] or [obsolete] or both. It isn't a normal or acceptable word in modern AE to the best of my knowledge. Most of the hits my web search brought up looked like obvious mistakes. A couple were, well, borderline plausible, contrasting, say, 'straightly' with 'crookedly.' The rest were archaisms matching Paul's town-crier sense, e.g.:
hath caused certain good ordinances to be devised and accorded by her Council; which also her Majesty's pleasure is to be well regarded, straightly kept, and...
ON the Queene our most gracious and most benigne soueraigne Ladies behalfe, we most straightly charge and commaunde you, that ye the sayd Aldermen, faile not
governor of his royal person and protector of his realms, dominions, and subjects, and the rest of his majestys Privy Council, straightly willeth, chargeth...
and for the settling of a religious and peaceable government both of the Church and State, do by these our special Letters straightly charge...
"Answer me straightly now." "Straightly I answer thee..."
This sense (= strictly, rigorously) is more often rendered as 'straitly', e.g. in the King James Bible, where the phrase 'he straitly charged them' most often corresponds to Luther's 'er bedräuete sie' (Mt. 9:30, Mk. 3:12, Lk. 9:21, Acts 4:17; except Acts 5:28: 'mit Ernst geboten'); or in Shakespeare, where Schlegel translates 'straitly commanded' as 'streng befohlen' and 'scharf mir untersagt' (Richard III: I.i, IV.i).
In any case, as far as I can tell, there's no difference in AE/BE usage, since dictionaries may list 'straightly' as a variant in both regions, but native speakers so far have come up empty. If any forum users do use the word themselves, or have seen or heard it used by reputable speakers, let us straitly charge them to cite examples forthwith or forever hold their peace! *g*
At the very least, the normal form 'straight' should surely be added with the marking [adv.], since it's an irregularly formed adverb.