“I think I can prove my disability commenced years ago, long before I lost my work credits and when I was still insured for SSDI. If I apply and allege disability during a period when I was still insured, will they forward my claim to DDS for the medical development and let me try to prove my allegations?”
Yes, this is no problem. You must simply accurately answer the questions on your application, especially the ones pertaining to when you became unable to work. They will develop your claim with that old date in mind, not just today. The main negative for you is that if you lose your claim (after exhausting all appeals), you will not have an opportunity to apply again for Social Security Disability, unless you again gain sufficient recent credits from work.
Also, if you are "poor" enough, you can apply for SSI as well (or instead) - as that does not require proving disability when insured - you need never have been insured at all for Social Security Disability, actually, to be eligible to file for SSI. As such, current strong evidence, easier to get, can suffice and the past evidence won't much matter for an SSI claim.