Ref continuing the climb, the SFF checklist may well fix the problem, if you come steaming down while running the checklist and subsequently fix the issue you’ll need to divert due to lack of fuel and you’ll look a bit silly explaining that to atc, the pax and management.
The best practice is to calmly run the SFF checklist, then if necessary calmly do some decision making and then if necessary calmly initiate a diversion.
Hetting into a panicky rush reduces safety and helps no one.