If you forget the declaration of intent before Barukh she'amar, must you start over from Ma Tovu? If you forget the declaration of intent before waving the lulav, can you still burn the lulav Erev Pesach? If you forget the declaration of intent before counting the Omer, does your Omer count still count?
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Of course you fulfilled the mitzvah. Mitzvos require intent, not a declaration of intent! If you had intent, and just forgot the declaration of course you are yotzai. (See Nodeh BeYehuda YD 93 where it's clear a declaration is not a necessity)
Even if you didn't have explicit intent while doing it, why else were you waving the lulav, counting the Omer etc? Of course you had intent.
There is a story with a gadol who was mesader kiddushin, and the chosson was concerned after the Chuppah that he had not had intent for kiddushin (apparently despite saying harai at...), to which the gadol asked him, what else were you doing under there?
As for your question about burning the lulav, you can burn anything you want, there's no requirement for it to be a lulav shel mitzvah.
(There are no sources for some of the statements made in my answer, because they are just obvious conclusions from learning the sugyos).