One way to find out the correct way to pronounce the name of a city is to listen to the local population. Ten or twenty years ago, that might have meant a long trip, but, with YouTube, you can usually find something from the comfort of your own home.
Try this link, for example. (You only hear the city's name mentioned once, near the beginning on the story.)
One other thing worth mentioning here is that "correct" pronunciations are sometimes not exactly fixed. There may be a "standard" pronunciation, but local accents and dialects as well as personal speech patterns may affect an individual's pronunciation. Sometimes a vowel gets pronounced "lazily", particularly when the following syllable makes a word's preceding syllable hard to enunciate.
It doesn't surprise me that, if you listen very closely, you may hear some slight variations in first syllable of Pawtucket, ranging from an unmistakeable "aw" as in raw to a more relaxed "uh" as in putty. That's just how people talk. In a similar way, you might hear some variations in the last syllable of Washington or London, ranging somewhere between how one might pronounce tin or ten or ton, or din or den or done. It's not uncommon for a syllable to have a small range of acceptable pronunciations, even if dictionaries only list one.