Correct spelling, explanation: choosing is present participle/gerund of the infinitive to choose. It’s spelled with the double -o vowel. The most common way of creating this form is adding the suffix -ing to the base (with an omission of e- vowel). Therefore, choosing is the correct form.

Definition of choosing:
1. noun, picking out someone or something
I don’t like choosing what I want to eat.
Choosing new clothes shouldn’t be that hard!
2. verb (the present participle), deciding what one wants from two or more things
From all the men in the world, I’m choosing you.
Choosing one life partner is difficult.

Collocations with choosing:
Some most commonly used collocations include: choosing carefully, choosing freely, choosing between, choosing from, and picking and choosing.
Choosing words carefully is an important thing for Adam.
They want to have the possibility of choosing freely.
Sorry I’m late, I was choosing between a T-shirt and a dress.
Choosing from this rich menu is tough.
Rules are not for your picking and choosing!

Incorrect spelling, explanation: since we have to add the suffix -ing to the core word if we want to create a present participle, the right form is choosing, not chosing. This mistake may come from the fact that the past tense of choose is chose.