I get a little giddy whenever constructors hide a cheeky little color joke in a puzzle, and 'embarrassed' is one of those go-to surface words that screams "RED" to me. In many themed American-style puzzles the setter will use 'embarrassed' as a hint that the letters R-E-D are hiding somewhere inside the theme entries (sometimes spanning a word break), or that a phrase has been altered by inserting a color-related chunk like RED, BLUSH, or
Flush. So you often see theme entries that are ordinary phrases containing the substring 'red', like 'PREDICAMENT' (P-RED-icament), 'SCAREDY-CAT' (scaREDy-cat), 'HUNDREDTH' (hunD-RED-th) or 'INFRARED' — each of these can be clued with playful surface text about shame or flushing.
Another common device is to make wacky theme answers by dropping or adding RED to familiar phrases. For example, a base phrase like 'PLAY
Mate' could be reimagined as 'PLAY-RED-MATE' in a gag theme, or constructors might hide RED across two words so that the clue 'embarrassed' points solvers to the hidden substring. Beyond the literal 'red' trick, synonyms like 'blush', 'flushed', 'red-faced', 'abashed', 'ashamed', and 'mortified' are used as straight definitions or as indicators for other types of wordplay (anagrams or containment). When I'm solving I look for those substrings and for color words crossing word boundaries — that's usually where the theme entries live. I love it when a simple clue-word like 'embarrassed' doubles as both a definition and a mechanical pointer to the theme; it feels clever and satisfying.