Yes, we can disabled the triggers.
You can almost certainly disable one trigger from another using an EXECUTE IMMEDIATE statement:
EXECUTE IMMEDIATE 'ALTER TRIGGER trigger_name_here DISABLE';
However, you definitely shouldn't be using triggers for application logic. It's a messy business, not least due to the fact that triggers aren't guaranteed to fire in order, but also because of the kind of "problem" you're experiencing.
It would be much easier and significantly safer to move all of the functionality you described to a stored procedure or package, and use triggers only where necessary for validation purposes.