Did I use correct polish off?

More or less.

I didn't eat lunch and because of that I was starving to death, so I ate the cake and meal and polished off my dish.

I would have said "cleaned up my plate", but polished off would be understood.

I didn't eat lunch. Because of that, I was starving to death and so I ate cake and then a meal and polished off my plate.

I'm assuming you ate cake before the meal. Usually cake is a part of a meal, eaten at the end.

Thx so much

I don't know about the US, but in British English the informal idiomatic phrasal verb 'polish off' is not used quite like that. The verb means to quickly finish or consume something. In an eating or drinking context, one polishes off food or drink, so you wouldn't say 'I polished off my plate' unless you actually meant that you ate the plate itself. You polished off the food on the plate, or you cleaned the plate as has been noted. Also you can use the phrase when you completed a task or job quickly. I polished off my geography assignment.