Just try your luck and spam applications, my friend. As mentioned, it really depends. Some HR may check some may not (but expect ministries to be super stringent). Finding a gov job w your qualifications is not entirely impossible, but with private 3rd class you usually have to settle for lower starting pay in gov jobs, based on the pegged rates.

Don't be disheartened, yeah. Just apply and see where it gets you. Even if you have to start at a lower tier just take the job, as a start somewhere is always better than going nowhere.