UPDATE: 10 of the words are four letters and 2 are 3 letters long. This reduces my list to 126 words, which are:

act age aim atom aunt auto cage camp can case cash cat chat chef coin come cost cup cute east echo egg face fame fan fat fee fine fish fit foam fog foot fun gain game gap gas gasp gate gift goat gun hat hen high hint hip home hope host huge hunt ice icon inch into item join jump just main man mass math mean meat menu mesh miss moon much must name nest net nice nose note nut off once one open pact page pass path pen pet pig pigeon pipe poem poet post push put quit safe same scan sea seat ship shoe shop sign sing snap sniff soap soft song soon soup spin spot stem step such suit sun unit upon use