Put some spring in your elbows when you hold hands with your partner. Proper arm tension will help you respond to your partner's movements. If your arms are too limp, you won't be able to respond to your partner's cues.
Maintain several points of contact with your partner. The hands are important, but the educational site Round Dancing says that partners can communicate more fully if they use other points of contact, such as the lead's hand on the follow's back, the follow's hand on the lead's shoulder, contact through the hips and torso, and eye contact.
Start out with basic steps until you get used to your partner. Everyone dances differently, and just because you did very well with your previous partner doesn't mean your experience will be the same with all partners. Stick to the basics at first so you can get a feel for how your partner dances.
Don't lead or follow blindly. It's true that each partner has a different role in swing dancing, but forcing a follow to dance the way you want is not good partner dancing. A follow also shouldn't wait for the lead to make all the moves. Partner dancing is similar to having a conversation---and if only one person is doing the dancing, it is a one-sided conversation.
Dance with many partners to get a feel for how other people dance. Even if you came to a swing dance on a date, it is perfectly acceptable to dance with other people---swing is, after all, a social dance. Dancing with many people will make you more receptive to cues from a new partner.