I'm not sure I fully understand the details of your issues, but I will do my best to try to help.
If you cancel your Internet Service Provider service, Cox, then you will have NO internet access for anything (neither email, nor webmail, nor dish/cable TV, nor web browsing, nor any other service on the internet), other than through your mobile phone's carrier.
You will need to find another ISP if you want to access the internet from your computers and other devices, unless you are going to route everything through your mobile network.
So, if you cancel your ISP service, you will not be able to access gmail (or anything else on the internet) from ANY of your devices at home, because you will have no internet access (other than through your mobile carrier).
As @Trikein tried to explain, you do not have to have or use an email account from Cox (or any other ISP) in order to access gmail (or a similar webmail service). You only need to set up an account at gmail (such as firstname.lastname@example.org) AND have access to the internet. That is the bare minimum (because one can also use an email client to access gmail).
However, for most ISPs, one does need to maintain at least one ISP account (such as email@example.com), so that they can communicate with you about your account and other important information. That does not mean that you cannot have or that you cannot access your gmail. The ISP mail account (XXX@cox.net) and the gmail account (XXX@gmail.com) are entirely unrelated. You can have both, even if you prefer to use gmail for your routine email communication.
Chances are, all that needs to be done is to correctly configure your various ISP email and webmail/gmail accounts, using the proper incoming (POP3 or IMAP) and outgoing (SMTP) server settings for each account.
If Cox is like some other ISPs, you will NOT be able to route your gmail through their ISP servers.
You will be limited either to accessing gmail from its website, and/or via some sort of email client that has been set up with the correct gmail server settings (as @Trikein explained earlier).
- If you "cancel Cox", then you will have no access to the internet AT ALL from your computers or other devices, other than what you might be able to access on your mobile phone's network (Verizon, Sprint, AT&T, T-mobile, etc.).
- And if you use Cox for your TV (and/or digital phone), too, you will also lose that.
- So, I'm not sure you really want to do that, unless you plan to sign up for a different ISP to provide those services?
- You will need some kind of internet access (via either your ISP or your mobile phone network or both) in order to access the internet and to access your gmail (or any other email or webmail service).
I hope this helps a bit,