Is there a technical name for a word that contains other words? I'm thinking about the contained words having all the letters in the order they appear in the original word. For example, "spawn" contains spa, pawn, paw. I'm also hoping to find a more technical word than "Kangaroo words" or "Matryoshka words" if there is one--something like "paraprosdokian" or "palindrome" etc.
Kangaroo words may be a special case of what you describe. They are words that contain embedded words (possibly disjointed) which are roughly synonymous with the larger, host word. For example, "observe" contains "see", albeit disjointed (here).
I know of no technical term for the more general phenomenon, but you might call them Matryoshka words, as is done both here and in the first link. It's suggestive of what you want and colorful.
This is a duplicate of a closed question referring to "embedded words." The top ranked answer was subsequence from the mathematics. However, the natural transition to subword is also possible. It is a mathematical term [Wiktionary] meaning "substring" which is more particular for contiguous symbols.