The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
SURF, DASH, SCORE, and KEY all fit the category "Words before board". Put one word after each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Kitchen verbs
SEAR / DICE / FOLD / WHISK
SEAR, DICE, FOLD, and WHISK all fit the category "Kitchen verbs". These are actions from a recipe. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things that bloom
TULIP / ROSE / FIREWORK / ALGAE
TULIP, ROSE, FIREWORK, and ALGAE all fit the category "Things that bloom". Each can bloom in its own way. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Weather events
FOG / HAIL / DRIZZLE / SQUALL
FOG, HAIL, DRIZZLE, and SQUALL all fit the category "Weather events". Forecasts warn you about these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.