With reference to cursive writing, the phrase to dot the i’s and cross the t’s, and its variants, mean: to pay attention to every detail, especially when finishing off a task or undertaking; to be accurate and precise.

The earliest occurrences that I have found of the phrase to dot the i’s and cross the t’s and variants—used literally and figuratively—are as follows, in chronological order:

1-: From the transcript of a debate on the union of Maine and Missouri, which took place in the U.S. Senate on Friday 14^th January 1820, published in the Daily National Intelligencer (Washington, District of Columbia, USA) of Tuesday 1^st February 1820:
—In the following, James Barbour (1775-1842), Senator from Virginia, was replying to James Burrill (1772-1820), Senator from Rhode Island:

It is next objected by the gentleman from Rhode Island, that the committee have exceeded their powers, in recommending this amendment. Pray, sir, what is the object of referring a bill to a committee—merely to dot the i’s and cross the t’s? I had supposed they had a more important duty to perform.—Not only their right, but that it was their bounden duty to modify or amend any and every part in relation to the particular subject embraced by the bill, and to extend its provisions so as to embrace every corresponding subject.

2-: From Harry and Lucy Concluded; Being the Last Part of Early Lessons (Boston: Munroe & Francis, 1825), by the Anglo-Irish novelist Maria Edgeworth (1768-1849):

“You make three kinds of r’s, and when I have learned to know one of them, then comes the other, quite different; and all your m’s, and n’s, and u’s, and v’s, are so alike, no human creature in a hurry can tell them asunder; and you never cross your t’s, so how can I tell them from l’s.”
“But I do dot my i’s,” said Harry.
“Yes, you do; but you never put the dots over the right letter; I can never guess to what heads the hats belong.”

3-: From a letter dated New York City, January 1826, that a person signing themself Monticola wrote to one Gaylove Cameron, assistant teacher at a grammar school in Vermont, published in the New-York Mirror, and Ladies’ Literary Gazette (New York City, New York, USA) of Saturday 21^st January 1826:

I have found in Mr. George P. Morris a very pleasant and good humoured acquaintance, who has promised to cross all the t’s and dot all the i’s, and to publish the letters in his paper.

4-: From A key to the new system of orthography, appended to The Fourth Epistle of Peter. Translated from the Original (New York: Published at the office of the Olive Branch and Christian Inquirer, 1829), by the U.S. evangelist and theologian Abner Kneeland (1774-1844):

There is no occasion of accenting the vowels in writing, as all writing will be perfectly understood without; nor is it necessary even to dot an i, or cross a t. But, for the sake of foreigners and children, these things are necessary in print.

5-: From the transcript of a speech that Littleton Waller Tazewell (1774-1860) delivered on Monday 16^th November 1829 during the Virginia Constitutional Convention, held in Richmond, Virginia—as published in Proceedings and Debates of the Virginia State Convention, of 1829-30 (Richmond: Printed by Samuel Shepherd & Co., for Ritchie & Cook, 1830):

Does not our own experience too, inform us, that a Senate consisting of twenty-four members, sitting up-stairs, can never restrain the power of a House of Delegates consisting of one hundred and twenty members, sitting here? The Senate may sometimes prevent the hasty and incorrect legislation of the House of Delegates; they may dot the i’s or cross the t’s, or correct the orthography in bills which have passed the House, (if it be allowable to suppose that any member of that body may not know how to spell,) but it never has and never can arrest any deliberate measure which the House is disposed to persist in.

6-: From the transcript of a speech that Philip Doddridge (1773-1832) delivered on Tuesday 24^th November 1829, during the Virginia Constitutional Convention, held in Richmond, Virginia—as published in Proceedings and Debates of the Virginia State Convention, of 1829-30 (Richmond: Printed by Samuel Shepherd & Co., for Ritchie & Cook, 1830):

The Senate has heretofore been what it was intended to be—a body of calm, reflecting men, not disturbed by any agitation originating with themselves, but having time to regulate and check those of the other branch—having in fact a much more elevated and useful duty to perform than merely to dot the i’s and cross the t’s of the other body. The proportion of bills from the House of Delegates which had been rejected in the Senate had been large.

7-: From Three Courses and a Dessert (London: Vizetelly, Branston and Co., 1830), by the British author William Clarke (1800-1838):

Impressed with the force of his own arguments, the purveyor of the preceding courses has attempted an epilogue to his entertainment; in which, he trusts that he has not presumed too much on the usual leniency of after-dinner criticism; and that none of his guests are of the delightful class of censors, who flourish a flail to demolish a cobweb,—who indulge in proving, by very elaborate and profound arguments, that there is but little substance in “trifles light as air;” or who occasionally go so far, in fits of ultra fastidiousness, as to cross an author’s t, and dot an i for him.

8-: From Business for the reformed Parliament, published in The Atlas (London, England) of Sunday 9^th December 1832:
—Note: in the original text, the i of is and the t of ts are in italics:

Mr. Hume is to bring forward 279 bills, and 11,348 resolutions on finance, including 2000 tables of expenditure, and an estimate of probable savings in government offices, with other calculations, including, it is said, nearly ten millions of billions of figures. It is understood that an important reduction will be effected in the public departments in the consumption of ink, by compelling the clerks not to dot their is or cross their ts.