Ocean and Puget Sound sailor here. The answer is: it categorically depends. If motoring into the wind in irons (in line with the wind), douse the sails. You're not getting any thrust when in irons. You could damage something or put in undue wear and tear.

But if you're asking about tacking, using the engine can drastically improve the point of sail. And if you're rigged to close haul, e.g. big genoa + Barber hauler, you can get very close to the wind.

With regards to your "little or no wind" context, as long as the sails aren't luffing, you're gaining some thrust from the sails in this situation.

But one measurement is worth a thousand opinions. Try it out for yourself. Watch your SOG and how close to the wind you can get with the transmission in forward vs neutral. You'll find there are sweet spots for your rig and sail configuration.