I am also having the same issue, with my 2018 IconX and now my Buds. Only Android messages is turned on for notification (full read out). Both issues still there when I upgraded from S9+ to S10 and IconX to Buds.
I tried Samsung's default messages, had same issue of double read out AND my Buds will keep reading out old short code text messages only and not the new incoming sms (I have my CC send me a text whenever there is a charge). Very frustrating to use with stock Samsung Messages, at least I can get new sms read out to me instead of just the old short code sms. Can't believe Samsung QA let this go thru.
Any suggestion besides clearing cache resetting Buds/phones (already tried) would be appreciated.