Need native check please.

This is okay grammatically.

However, I feel it sounds a little awkward. I would suggest シャワーを浴びてきます。or シャワーを浴びてくる。for "I'm going to take a shower".

Thanks !
And for « I'm going to eat » which one is the most natural ?

I'm going to take a shower.
The speaker is "I", usually "I" speak to someone. So シャワーを浴びてくる implies that "I" go to the bathroom, take a shower and go back here.
If the subject is she, he, they, Tom or Mary, then "彼女はシャワーを浴びにいく" is okay, it sounds like a description in a novel.

I would use シャワーを浴びにいく as a suggestion. For example, my family are on the beach. We spent good time there. Before leaving, I find a simple shower room near the sea. Then I ask my family "シャワーを浴びにいく？".

I'm going to eat.
My Japanese translation is just "食べるよ" or probably "食べるつもり".
If the sentence is like "I'm going to eat out." or "I'm going to eat lunch.", I would use 食べてくる. And if the subject is others like "Tom is going to eat lunch." I think トムは昼食を食べに行く is good, while トムは昼食を食べてくる sounds odd.

Very interesting! Thanks for the explanation! :)