These days, I couldn't keephaven't been able to keep up with my diary because I washave been really busy with my final exams and reports.

"These days" sets the narrative in the present, so you will need to use the simple present or the present perfect.

From today, I'll keep up with my diary as much as I can.

The cause of pfood poisoning iwas probably eatieng raw chicken meat.

To be more natural, I would rephrase to "the food poisoning was probably caused by."

"Chicken meat" is understandable and correct in certain situations, but here, "raw chicken" is more natural.

I had a fever, stomach ache, and diarrhea for likeabout three days.

"Like" is fine verbally but in writing, I would avoid it.

I was in a bad shape.

My mother says that my family easily get food poisoning byfrom eatieng raw meat, so I'll never eat raw meat again.

I would add "again" or something similar to the end since you did eat raw meat this time, whereas "I'll never eat raw meat" suggests that you never have at all.

These days, I couldn't keepstay on top of my diary because I was really busy with my final exams and reports.

You can "keep up with"/"stay on top of" a habit (like writing in a diary), you can "keep" an object in a location (like keeping your diary on your bookshelf), and you can "keep" doing something over and over (like, "I kept running despite the rain"). However, you can't mean "keeping up with" a habit while shortening the whole phrase to "keep"

From today, I'll keep up with my diary as much as I can.

The cause of pfood poisoning is probably eatieng raw chicken meat.

I was in a bad shape.

The phrase "in bad shape" loses its meaning if you had an article like "a". When you add the "a" like "In a bad shape", I think you mean a literal bad posture (your bent over in a painful position).

My mother says that my family easily gets food poisoning by eatieng raw meat, so I'll never eat raw meat.

These days, I couldn't keep up with my diary because I was really busy with my final exams and reports.

"Up with" makes it more natural but 'keep my diary' is not grammatically wrong.

From today, I'll keep up with my diary as much as I can.

More natural:
From now on, I'll keep up with my diary as much as I can.

The cause of pthe food poisoning iwas probably (eatiegng) raw chicken meat.

You can drop 'eating' if you want.

My mother says that my family easily gets food poisoning by eatieng raw meat, so I'll never eat raw meat.

More natural:
My mother says that my family is really prone to food poisoning from raw meat, so I'll never eat it again.