If you are expecting a windfall in July that will pay off what you owe, maybe you could just file for an extension and pay it all off at once instead of dealing with the payment plan?
Extensions do not delay payment, they just delay filing the forms.
Yes, but when you figure up the penalty and any interest for underpayment, one of the methods of figuring out what you should have paid is based upon your expected income tax--which uses your prior year as a baseline. Supposing for a moment that your withholding at least matched that for the prior year and your income (especially any from self-employment) didn't change much, the penalty and interest might not actually amount to much if you file an extension in April and really are in a position to pay the whole thing off in July.
And, of course, you'll want to check this with a CPA. My tax experience (as an assistant to a CPA while I was in grad school) was a few years ago, although I do still do ours.