So, with all this information online, I figured everything can be done nice and fast.
Not so. I called HIP and talked to someone friendly and helpful. She is, in her best helpful manner, going to put a paper application for health insurance in the mail, and I should have it in five business days. And no, it can't be retroactive to April 1: it'll start May 1. I'm going to call Aetna, even though they're $68/month more, on the off chance that they'll cover me for April. (Everyone else is a lot more expensive than either.)
Well, I was going to call Aetna. Their phone system is set up to deal only with existing customers and doctors. I might get around that, but it doesn't leave me thinking they'll be faster than HIP, so I'm not going to spend the extra. *sigh* Can I live in a civilized country, please?
On the positive side, I have just gotten another bottle of Neurontin, enough to get me through April, and I'm getting my eyes looked at tomorrow. Now I have to very carefully not get sick next month.