If the OP has no TSA Pre, then the educated assumption is the OP has no GE. GE comes with complementary Pre.

OP, BA will arrive in T7, while DL will leave in either T2 or T4, B6 will leave in T5. I would recommend booking the later flight with B6 to give us a few extra minutes. When you clear immigration and customs and are at landside at T7, look for airtrain sign. Make sure to take the on that circles around terminal. Do not take the one that goes to federal circle, Jamaica or Howard beach.

Also, are there AA flight to CHS, can you get them on one ticket with your BA flight? AA is also in a different terminal (8), and you need to transfer landside via airtrain.