I’ve been working as a part timer since I quit my job as a regular worker on Aug.25th
Today, I’ve done the procedures for my resignation.
Today, I've completed the procedures for my resignation. <--- native
First, I went to a public employment security office near my office to get the certification that proves I really quit my job.
When September begins, I’ll have to go to the office near my house
and go through another procedure that
extend extends or will extend the starting time in which I can get themy unemployment insurance.
【！】Although "the" is specific, in this case your sentence is directly related to you, so it should be possessive.
I start getting my unemployment checks next week. possessive
I will apply for my unemployment checks. possessive
I need to apply for unemployment insurance. not possessive.
Secondly, I headed to a pension office to secede from the pension system.
As I do these procedures, my resignation seems real to me.