You are correct in that your mode should be one of "Normal", "Scientific" or "Engineering". Both Engineering and Scientific display answers with an exponent. According to the TI-84 manual, "Normal" mode uses up to 10 digits in the answer with the decimal point somewhere in the middle. However, there is this note:
Note: If you select Normal notation, but the answer cannot display in 10 digits (or the absolute value is less than .001), the TI-84 expresses the answer in scientific notation.
You will notice that the input "2574/2598960" gives the result "9.903961585E-4".
But "2574/259896" gives the result ".0099039616", which is what you expect.
The simple answer is to write the answer down and move the decimal point 4 digits to the left and leave off the exponent.