Discussing shakai hoken in Japanese

Can someone translate this for me? I could easily use Google Translate, but I'd rather it be in natural Japanese to avoid any misunderstandings.

I am currently in the process of moving out of (city) and changing employers. I will soon be enrolled on my new employer's 社会保険. Can I withdraw from (city)'s 国民健康保険 and 国民年金 through the mail? It will be very difficult for me to visit (city) City Hall during weekday hours.

*** FYI I know it's very unlikely that they will allow me to submit this paperwork through the mail. However, they did allow me to submit other paperwork through the mail... So here's to hoping! It would be much easier than taking a new day off so early from my new job. The start date has been moved up due to the woman I'm replacing leaving very suddenly.

now: you are in city A. you are in 国民健康保険 and 国民年金 of city A. you have 年金手帳.

moving from city A to city B.
changing from company A (in city A) to company B (in city B)
future: you will be in city B. you will join the company's 社会保険.

moving from city A to city B.
you have to go to city A office and write a document A,転出届, which shows you have a moving plan from city A to city B. you will receive the document B,転出証明書, from city A.

after you move to city B, you have to go to city B office within 2 weeks. you give them the document B,転出証明書,(which you received from city A) and write a document C, 転入届. they will write your new address on 在留カード and マイナンバー通知書（カード）.

at the both times, you had better bring your passport, your 在留カード, マイナンバー通知書(カード)
, and 国民健康保険証 (if you have).

after you finish the paperwork at city B, city B inform city A that you actually have moved (and you finish the paperwork). then, city A will write your moving on their database that you move from address A in city A to address B in city B. if you don't take the necessary paperwork at city B, your moving will not be recorded on city A's database. (pending)
(if you don't take the paperwork at city A, you can't take the paperwork at city B. you will be refused, because of the absence of 転出証明書(B).)

at the new company B, after you fix the new address(B)
you show your 年金手帳 and fill some documents. company B will take care the following paperwork that joins you 社会保険. the organization of company B's 社会保険 (generally, 日本保険機構) will inform city B that you join 社会保険, instead of you. ( you don't need to any paperwork to remove 国民健康保険 and 国民年金.)
日本保険機構 have all your record of 国民年金 and 社会保険.
社会保険 overcomes 国民年金.( it means that ,if you join 社会保険, 　国民年金 and 国民健康保険 will be automatically cancelled).
but, if you finish the joining of 社会保険(when you change the companies), you have to take paperwork to join either another company's 社会保険 or 国民年金(and 国民健康保険) .

some paperwork can be done by post mail. but, it takes time and also you have to fill all necessary columns with no mistake. it is better to go to city offices.

Hello Ken, thank you for your very thorough reply! Let me see if I understand... I MUST enroll in my new city's National Health Insurance. Once my new company enrolls me in their shakai hoken, National Health Insurance will AUTOMATICALLY be cancelled? I don't have to do anything and it's automatically cancelled?

Sorry, I just want to make sure because my friend said even after joining shakai hoken the city continued to send new National Health Insurance bills - so he was paying for TWO health insurances!

sorry, I made mistakes.
the system is different between pension and health insurance. (I made a confusion.)
about pension: you don't have to do anything, because national pension and company's pension (shakai hoken) are totally controlled at 日本保険機構. they know the double joining. they will tell you the refund automatically.

about health insurance: company's health insurance (shakai hoken) and national health insurance are independent.
company's health insurance is under 日本保険機構, whereas national health insurance is under city office.
you have to take paperwork at city office, when you change health insurance, from national health insurance to company's health insurance.
since one person can join only one health insurance, if you pay double, you have to ask the refund.
in that case, you can ask the refund of national health insurance.

"I MUST enroll in my new city's National Health Insurance."
you don't need to do that, if you are very sure that you will join the company's shakai hoken.
in April, many freshmen (also women) will move and get jobs in Japan. they will not join national health insurance, when they are sure they will join their company's shakai hoken. that is quite ordinary.