Used the (pronunciation)
"...what I heard instead of 'used the' was like 'use the' "
If we are speaking clearly and enunciating every word carefully, there would be a clear /d/ as a final phoneme in "used".
However, the following word "the" begins with the phoneme /ð/. This is closely related to /d/ in production.
/d/ is alveolar, but /ð/ is an inter-dental fricative.
i.e. The difference is the tongue touches the palate and behind the teeth for /d/, but the tongue protrudes more for /ð/ (interdental) and is vibrated (fricative) .
The difference is often missed when speaking quickly. The first phoneme is usually elided by most native-speakers when the tongue of the speaker passes from the first phoneme to the second.
It will sound like "use the" i.e. /juːz/ /ðə/...
not "used the" i.e. /juːzd/ /ðə/
I have always called this liaising but there are probably other terms.
By way of comparison, we also have
used to
In this case "used" is liaised with "to".
The "to" begins with the dental phoneme /t/, the preceding dental is also reduced, and elided.