In ‘Unabdil Enbadu’ upanyasam series, we were told that we should not partake anything without offering it to Emperuman? What about pickles and appalam? How to offer them?

Yes, one should not partake anything without offering it to Perumal. When we make pickles, we offer the whole batch the day we make them; similarly, when we make appalam dough, we offer the appalam balls made out of appalam dough to Perumal.
Moreover, our elders have said we can make a fresh batch of appalam and offer it on the same day as it is not considered tainted. Elders have the practice of offering fried appalam from fresh batch on festive days.