ごめん！受験勉強で年賀状送れなかったって英語でなんて言うの？

These are both polite sentences to apologise for your business. It is nice to start with a sorry, and then to give your reasons. When something slips our mind we mean that we have been preoccupied with other things so it makes us a bit forgetful. 'The thought slipped me mind' - it left your mind.

If something takes priority over something else it means that it must be completed first and receive all your attention for the time being. Less important projects must wait ( like sending New Year cards).

Sometimes everything happens at once and we have to skip some activities so that we have time to complete the most important activities. Usually, our studies are a high priority. The easiest way to apologise is to simply say "sorry". If we wish to be more formal, we can say "I apologise for ..." or "Forgive me for...". Usually we would say "Forgive me" if we did something terrible, much worse than not sending a card!

When you add a "so" to sorry, it shows that you are truly sorry and that you feel bad. The rest of this sentence explains that you were unable to send the card because of your studies.

To apologize for not sending a New Year card to your friend, you can say: "I'm sorry I didn't send you a card, I was studying for my exams." This way, they will know you are sorry and know why you were not able to send one at the same time.

School can get really busy and we might not get to have all our plans play out. We might not be able to visit friends or to have guests over. In this case, forgetting to send a New Year card because of exam preparation can be excused by using the sentences above. "I am so sorry I didn't send a New Year card this year, I was so swamped with school work." "Revision for the entrance exam was intense, so sorry I didn't send through your New Year card like I usually do." Swamped: overwhelmed, drowning under something (work)