You did not answer some of Trikein
's questions. I have them repeated below.
Where are your emails now for Inbox.com? Do you go to their website to check their mail or use a program like Outlook? If Outlook, do you know if you are using POP or IMAP? If you want to know how to tell, reply with what email program you use and I can include steps on how to check. One easy way to tell is if you delete a email from one device(PC), and it automatically deletes from another device (cell phone) then you are using IMAP.
If you use an email client like Outlook or Thunderbird to access your Inbox.com account and access it by way of POP3, then you will not lose any of your folders are those message should be stored locally on your computer. As long as your do not fully delete that account from the email account, but rather just disable the account (i.e. prevent it from check for new mail), then those message will go no where.
If you are using an email client like Outlook or Thunderbird to access your Inbox.com account and access it by way of POP3, then you can just copy those folders either over to local only folders (how might very depending on the email client) or over to another IMAP email account. In the case of the later, no new account is needed. In the case of the former, the emails will then be uploaded to the servers of that new IMAP account while also remaining on your computer (as long as you do not delete them).
As to free email services, find one that you like. Your current Gmail account will certainly do the trick. Or you can go with something else.