Use a short list and move with intention
You are trying to decide how picky to be when you tour homes versus how fast you need to move. My guidance focus your touring list around the price bands that are actually being chosen, and plan for a multi-week process instead of a weekend sprint in Raleigh, NC.
Here is the constraint I plan around based on the previous 30 days new listings entered the market at a typical asking price of $440,000 last month in Raleigh, NC, and there were 713 new listings. This changes your plan because a wide-open search tends to waste your best decision energy on homes you were never going to buy. With a typical new-listing ask of $440,000 and a typical sold price of $400,000 last month, you can build a realistic filter that keeps your tours aligned with what you can win without stretching. Some metrics were not reported for this period. Filter your tour list by price first using the $440,000 typical new-listing ask as a reference, then narrow again based on the home type you will actually live in long term. Keep your calendar honest a typical closed sale took 41 days last month, so line up your lender, decision-makers, and showing schedule so you can act without rushing. When you find a fit, do not stall out waiting for a perfect alternative, because supply was 2.67 months recently and that favors decisive shoppers.