Is it just me, or is the letter "i" pronounced differently in the words die and dice? In die, it sounds like the regular long I sound (ay), but in the word dice, it sounds closer to something like "əy" with a short schwa sound.
You’re right: one is [aɪ] but the other one, which is shorter and higher, is [ʌɪ].
This phenomenon is called, dubiously at best, Canadian raising. Most North Americans do this with that diphthong. Wikipedia writes:
Raising of just /aɪ/ is found throughout the United States, and so may be considered an increasingly common General American characteristic, with the only major exception in the U.S. being in the South.
It happens mostly before voiceless consonants, of which the /s/ at the end of dice is one. That makes dice come out as [dʌɪs] but die is just the unraised [daɪ]. Other pairs are tight/tide, writer/rider, and the two versions of high school.