I want to get better atimprove my English skill and cooking skills.

The other corrections are good, but this is fine as well

When I negotiate something with someone and try dealing with a customer’s complaints, I would struggle with how to convince and how to say with formal and correct English.

In addition, I don’t like speaking on the phone in English, i. It’s too difficult to listenhear to what they say.

I prefer texting to calling but I know it's something I have to practice it.

Your original sentence makes sense, but the 'it' at the end is a bit unnatural.

I want to get better at English skill and cooking.

I’ve lived in Australia for five months but I don’t feel my English skill improving, especially listening and speaking.

Needless to say, I’ve been on a plateau.

When I would negotiate something with someone and try dealing with a customer’s complaints, I would struggle with how to convince them and how to say things with formal and correct English.

In addition, I don’t like using the phone in English, -- it’s too difficult to listen tohear what they say.

I prefer texting to calling but I know I have to practice it.

For cooking, I can’t cook many kingds of food, so I want to step up my game and broaden my range of cooking.

As I love Thai and Indian foods, I want to try to make ithem from the¶
scratch by myself.

It seems to be difficult but if I could cook it, I could eat it everyall the time, -- it mustwould be amazing!

I want to get better at my English skills and cooking.

Since you are talking about your English skills directly, you should put "my" in front of it
Also, there's more than one skill involved in English (for example: reading, writing, listening, speaking...), you should say "English skills" instead of "English skill"

I’ve lived in Australia for five months, but I don’t feel like my English skills have been improving, especially with my listening and speaking abilities.

Sounds smoother like this.
You need a comma before "especially" and "but"

Explanation with especially: https://ell.stackexchange.com/questions/101127/should-we-always-use-a-comma-before-especially

Explanation with but:
https://www.grammarly.com/blog/comma-before-but/

Needless to say, I’ve been onhit a plateau.

When I negotiate something with someone and try dealing with customer’s complaints, I wouldtypically struggle how towith convinceing and how to say withtalk formally and correctly in English.

You can just say "customer complaints" when you are dealing in customer service, but if you are talking about one specific customer, then say:
When I negotiate something with someone and try dealing with a customer's complaints...

In addition, I don’t like to phone in English, i. It’s too difficult to listen to what they say.

I think it's Australian English to say "to phone", but in American English, I would say "I don't like to talk on the phone in English" or "I don't like to call in English"

I prefer texting to calling, but I know I have to practice ithis skill.

For cooking, I can’t cook many kingds of food, so I want to step up my game and broaden my range of cooking.

As I love Thai and Indian foods, I want to try to make it from the scratch by myself.

It seems to be difficult, but if I could cook it, I could eat it every time, it mustthese types of foods, I could eat it all the time and it would be amazing!