Would Putin have avoided most of the sanctions, if he had limited his invasion to the Luhansk and Donezk regions, instead of invading the entirety of Ukraine?

Probably. My guess is it would have played out similar to the annexation of Crimea.

Ukraine made it quite clear that it would not cede the regions under any circumstances.

Probably not. Would Germany avoid military conflict if Poland invaded the Brandenburg state and limited it only that? Would the US avoid military conflict if China took over California but promised to stop at that?