richdom is a rare obsolete word inherited from German. Don't use it. But when it was in use, the primary meaning was "Royal power, sovereignty" ("kingdom" was an even rarer sense of an already rare word). The only current English words I can identify that differ only by suffix -ness or -dom are heathenness and heathendom.
According to the full OED, both can be used as nouns, but only heathenness can be used as an adjective (which I confess I don't understand).
Both suffixes are still "productive", but note that -ness normally only forms abstract "nouns of state" (kindness, laziness = the states of being kind or lazy), whereas -dom (which also generates such terms) can additionally be used to form words with a (figurative) sense of ‘domain, realm’.
Hence the nonce-word Cockneydom would mean "the land where the Cockneys live", as opposed to Cockneyness, which would mean "the state / quality of being a Cockney".