I'm a Java developer. My exit from my previous employer was rather "unceremonious." I don't have ANY proof of my involvement with them, not even payslips. I destroyed it all. Call it self-sabotage. I quit after enduring "a prolonged emotional trauma." When I couldn't take it anymore, I called up HR & informed them that I won't be attending work anymore. After they gave up on trying to convince me to come back in, they convinced me to serve my notice period from home. The notice period went flawlessly, but since I never went back there (& never will), I don't have the relieving letter either. This was my 1st job after college, of which I only completed 3 (outta 4) years. How do I explain all this to my interviewers, who won't hire me without my previous employer's payslips, relieving letter, etc.?
I don't know what happened but based on the fact that you don't even want to call or e-mail them, I have to assume it was dire (even if it was only that way for you).
As stated in the comments, bank statements showing you were paid by your employer should be a sufficient replacement for pay stubs.
As for the letter, if you really need to avoid direct contact, do this through a trusted friend or family member (TFF).
1) Set up a really strict e-mail rule to send any email from your employer to a special folder. Have your TFF do it for you if necessary.
2) Have your TFF write the email requesting the relieving letter from your employer.
3) Any conversation about getting the letter gets filtered through your TFF. If there are certain topics or issues that might come up that you don't want to hear about, instruct your TFF not to tell you about it. Not only what was said but that it was in an email at all. You only hear what's strictly necessary to handle the logistics of getting the letter.
4) Once you have the letter, change your e-mail rule to delete any e-mail from your previous employer. Delete the entire folder of e-mails you've already received.
I would suggest using a throwaway e-mail address but I don't think your employer would necessarily trust a message from an address they don't recognize. You could also do the same thing via a snail mail letter with your TFF reading and writing the correspondence.
I don't see a way around either getting the letter or explaining why you won't get the letter. I'd be one thing if you couldn't get the letter, but I think an employer would view you not having it and being vague about why very poorly.