Reset your delimiter to something not embedded in your data.
$: echo "$line"
"DB-ConnString","Server=tcp:database.windows.net,1433;Initial Catalog=db;Persist Security Info=False;User ID=DATAFACTORY;MultipleActiveResultSets=False;Encrypt=True;TrustServerCertificate=False;Connection Timeout=30;Authentication='Identity';"
$: IFS="|" read -r -a my_array <<< "${line//\",\"/\"\|\"}" # change "," to "|"
$: printf "[%s]\n" "${my_array[@]}" # show the results
["DB-ConnString"]
["Server=tcp:database.windows.net,1433;Initial Catalog=db;Persist Security Info=False;User ID=DATAFACTORY;MultipleActiveResultSets=False;Encrypt=True;TrustServerCertificate=False;Connection Timeout=30;Authentication='Identity';"]