I'm also running out of time to write an English diary entry.

Like the other reviewer, writing an entire diary is not something you would do all at once. You could also say:

I'm also running out of time to start an English diary.

which also implies you are not completing the entire diary at once.

I have plentylots of of school projects, assignments and tests.

"plenty of" usually has positive meanings in English. Examples might be: I have plenty of food. I have plenty of friends. So because this is a negative thing to have a lot of, "lots of" is probably better.

I have too much to do and I don't know what to do about it.

I think the original could be left uncorrected, but it sounds a bit odd to repeat "to do" like that, so I added some words to make it sound a bit more natural.

I'm also running out of time to write an English diary entry.

"An English diary" would refer to writing the entire diary. So you can't really write one, because it is too much to write in one go.

So you can either say:

"Running out of time to write an English diary entry." Which means writing one individual post in the diary.

OR

"Running out of time to keep an English diary." The verb keep is the term used to refer to the act of maintaining a consistent habit of writing in your diary.