If believing doesn't make any difference in activities, is the belief true?

Assalamualaikum.

I am a by-born Muslim who has recently restored faith in Islam after deviating from the truth for a while. This question and fact are related to my own thoughts and philosophies and not directly linked to any of the messages of Prophet SM or the Quran. So, if you can enlighten me in this regard with the required links, then I'd be grateful.

My question is if we truly believe in something with our whole heart, then shouldn't we work accordingly and give the job our 100% effort? For example, if I told you that there is an exam tomorrow and if you fail that exam the teacher would get you expelled then I believe even the worst student (who wants to continue their study) would study 100% for the day. Similarly, if we truly belief with our heart that Allah, the Most Great, exists, and He would punish us (beyond our comprehension) if we do wrong deeds and don't worship regularly, then shouldn't all of us oblige by His commands? I mean sure, the punishments will come after an eternity of time, but they will arrive, and if we have, even the slightest of faith that even a part of the punishment will be given to us, we all (those who believe in Allah) should've deviated from the wrong.

So, my question is, if someone who proclaims to be Muslim still does forbidden activities, can his belief be called true? And if it can be called true then what other explanation there is (except having fault in his belief) that acknowledging the severe punishments, he/she is still doing it?

Thanks