Proper way of checking if row exists in table in PL/SQL block

I wouldn’t push regular code into an exception block. Just check whether any rows exist that meet your condition, and proceed from there:

declare
  any_rows_found number;
begin
  select count(*)
  into   any_rows_found
  from   my_table
  where  rownum = 1 and
         ... other conditions ...

  if any_rows_found = 1 then
    ...
  else
    ...
  end if;