Magento2 Varnish issues with customer session and site header

Hi

Magento version: C.E 2.3.5
Varnish: varnishd (varnish-4.1.1 revision 66bb824)

I am facing a very strange issue. In the admin panel,
Stores -> Configuration -> Advanced -> System -> Cache -> Varnish (recommended)

Now, varnish is enabled on my application.
1. I log into a customer account
2. Access some category pages, homepage, product pages. In my header, I see “Welcome <user>”.
3. I logout from the application

4. Now when the system automatically redirects to hompage (after 5 seconds of logout), in the header, I still see the dropdown with options “My acount”, “wish list”, logout.
5. If I access the already opened pages like, category pages, product pages, the same issue appears. 

Expected result:
In the header, the labels, “Sign In  or Create Account” should appear after logout.

Actual Result:
The system serves the cached version of the page, with a dropdown to logout for an already logged out user.

Did anyone faced any similar issues?