The types timestamp or timestamptz optionally take a precision modifier p: timestamp(p).
To round to full seconds, set the default to:
now()::timestamp(0)` or `now()::timestamptz(0)
Standard SQL functions CURRENT_TIMESTAMP (returns timestamptz) or LOCALTIMESTAMP (returns timestamp) allow the same precision modifier:
CURRENT_TIMESTAMP(0)
LOCALTIMESTAMP(0)
That's a bit cheaper and shorter than calling date_trunc() - which truncates fractional seconds (may be what you really want!)
Store timestamps as timestamptz (or timestamp), not using a character type.
Finally, to make sure that ...
newly added entries also have the format: YYYY-MM-DD HH24:MI:SS
you could define your column as type timestamptz(0). This will not only cover the default, but all values entered to that column.
See @Clodoaldo's answer for instructions on to_char() and how to ALTER TABLE.
This related answer for in-depth information on timestamps and time zone handling: