Yes, you are hearing correctly. You are right that the 'd' isn't pronounced in 'used to'. It is virtually impossible to pronounce a 'd' followed by a 't' in natural speech. The two sounds merge together, and only the 't' is said. You are also right that 'to' has two pronunciations: /tu:/ when it is stressed, and /tə/ when it is unstressed. In fact, the preposition 'to' is very rarely stressed. In the phrase 'used to xxx' it is unstressed, so the pronunciation is /tə/. The two words 'used to', spoken in connected speech, sound like /ju:stə/ . If the word 'Youstah' existed, it would be pronounced the same.

As commonly spoken, one can drop the 'd' when saying something like "I used to do that but I don't anymore". It is an example of connected speech that you can run into. However, it is not necessary to drop the 'd' What will make a difference in pronouncing this will be that you should pronounce the 's' in 'used to' as a 's' instead of like a 'z' as you might hear in the word 'used' as in "He bought a used car".

You are right, Addison, trust your ears. This a typical phonological feature of English, and can be observed in many situations. Here is another very common one: "Have to" pronounced /hæftə/. BTW, if the next word begins with a vowel, it's very common not to reduce the word "to" -- instead inserting a linking consonant, normally a "w." some British speakers keep the schwa and use a linking "r" instead of a "w."