Is it permissible for a servant to intend devotion to God with all their words and actions, and then strive to correct their intention for each act, or could this lead them into bidʻah (innovation in religion) or lying about God? Or is it better to be truthful with God by not intending to draw near to Him except through what is undoubtedly a pure act of drawing near for His sake?

If the intention is to inquire about the intention of drawing closer to Allah with every act that is neither forbidden nor disliked when performing it, along with striving against the self to correct the intention and make the work sincerely for Allah, then this is what is Islamically required and there is no innovation in it.

If the intention is to have sincerity in all future words and actions, and then to strive to correct and bring to mind the intention with each act, this is also valid. In matters of worship, if one is unable to perform an act they used to do due to an impediment (illness, travel, etc.) with the intention to continue, they are rewarded the full اجر (reward) for what they used to do. However, if it was not their habit to perform the act, they are rewarded only for the intention.

As for non-worship matters, it is probable that whoever intends sincerity to Allah in all their actions – whether pure worship or not, such as spending on one's dependents – and then performs something out of forgetfulness without intention, they are rewarded as if they had intended it. This is supported by the saying: "On your first day, intend that you do not perform except what pleases your Exalted God. If you combine this with the action, that is perfection. But if you do not, it is hoped that what preceded of good intention will suffice you, according to the scholars."