To dither and be undecided.
As is often the case with reduplicated word pairs, there is one word here that had a prior meaning while the other was added for effect. The usual sense of the question 'shall I' supplied the meaning, with 'shill I' the reduplication. The meaning is really just the same as 'shall I, shall I not?'
The phrase was originally to go shill-I shall-I or, to stand at shill-I shall-I. William Congreve's The way of the world, 1700, is the earliest record of the term in that original spelling:
"I don't stand shill I, shall I, then; if I say't, I'll do't."
The first record we have of it that uses our currently accepted spelling is from just a little later. Sir Richard Steele's The tender husband, or the accomplish'd fools, a comedy, 1703, includes:
"I'm for marrying her at once - Why should I stand shilly-shally, like a Country Bumpkin?"
See other reduplicated phrases.