I am unable to sell my primary residence (without giving it away) due to a market downturn. If I rent it and then buy a new house which will become my primary residence, can I take a hardship withdrawal to apply towards the purchase of my new primary residence?

It depends on what your plan documents say. Companies have some discretion in determining what qualifies as a hardship - but they must document the specific criteria. Some plans don't allow for any hardship distributions at all. So it's possible, but not certain, that your plan would define the purchase of a home as a hardship, even if it's not your first home purchase. Bottom line - contact your plan administrator and see what your 401(k) plan allows.