Learn Hardcore Turkish: Ben yarın sözleşmeye imza atacağım.

You can drop it. The verb ending already shows the subject. So Yarın sözleşmeye imza atacağım is perfectly natural. Keep Ben if you want emphasis or contrast (as in “I will sign it, not someone else”).

Why is it sözleşmeye with the dative suffix -e, not sözleşmeyi with the accusative?

Because the idiom is bir şeye imza atmak (“to put a signature onto something”), and it takes the dative (-e/-a) for the thing being signed. If you use the simple verb imzalamak (“to sign”), then you take the accusative: Yarın sözleşmeyi imzalayacağım.

What’s the difference between imza atmak and imzalamak?

Both mean “to sign,” but:

imza atmak is a light-verb expression (literally “to put/throw a signature”) and is very common in everyday speech.
imzalamak is the straightforward verb “to sign” and often feels a bit more direct or formal in writing. Both are correct with this meaning.

How is atacağım formed, and why is there a ğ?

Formation:

Verb stem: at- (“to throw/put” in this idiom)
Future tense: -ecek/-acak → last vowel is , so →