I've been to Molfetta and its an amazing place
As far as the language spoken there, everyone speaks standard italian. Last time i was there 1999, I was an anomaly since i was under 40 years old and spoke and understood the dialect (since most of my family came to the states pre-1960 they brought the dialect with them to America before it became uncivilized to speak it in italy
The cemetery in Molfetta is also amazing, but an immense place. Additionally, they have a system there (which i believe is standard in Italy) that removes the contents of graves after a long period of time and replaces the bones into a smaller ossuary. Unless you have relatives who died post 1920ish, you probably won't be able to find much. Although there is an older section of the "campasande" that has some older graves. Regardless if you find your relatves or not, walk around the place just to view the sculptures.
As far as records go check out this place
It is the diocese archive. I'm not sure how helpful or what you'll find there, but according to the website they have the older records that you may seek. I've sent letters to the churches themselves and have always received responses that they cannot fulfill my inquiry, although they do deny my request in a very friendly manner.
make sure you go inside la chiesa vechhia by the water (as well as melfette a'vecchie - the old walled part of the city" and also visit the cathedral to see the paintings done by rococo artist Corrado Giaquinto. A visit to Il Pulo (a neolithic site) is also essential, but make sure its open before you go.
sorry i couldn't be of more specific help, i no longer have family there and my contacts are limited