I find that a lot of people who have danced mostly the gent's role, when they start dancing the lady's role, think that when they swing, they're supposed to go backwards or sorta sideways... I mean, hell, a lot of people think that's how a lady swings when they first start dancing that role! But, I emphasize that you're both walking (or buzz-stepping) forward, roughly aiming at a point over your partner's shoulder. In fact, the footwork for a swing for both roles is exactly the same!!! A good way to emphasize this might be to have participants do a gender-neutral swing (e.g. right hand on shoulder blade, left hand clasped with partner above the heads) and then change the hand position into your classic ballroom swing (perhaps even trying out ballroom position with person A leading, then with person B leading) and noting how the footwork stays the same.
Ending the swing on the opposite side also takes some getting used to. Instead of emphasizing which role ends where ("gent ends on the left... oh god, am I the gent??"), I find it useful to talk about ending the swing with the "pointy end" pointing in the direction you want to be facing, and releasing from there--works for both roles, so no scrambling to remember what role you're dancing and what side you're supposed to end up on.
You might lean towards selecting dances where tricky gendered stuff (e.g. the courtesy turn, which many people will be learning for the first time in their new role) happens with a neighbor. If it happens that you and your partner are both confused about how to do a courtesy turn in your new role and the only courtesy turns happen with your partner, you're not going to get any help.
Lastly (and this is something that I urge allllll you folks on the listserv to do, even if you're not calling specifically gender-neutral dances): use gender-neutral language to describe the roles! I find this every effective for breaking down role gender expectations, even if the role names themselves are gendered. So rather than "gents, take that neighbor lady and scoop her around in a courtesy turn", say, "scoop
them around in a courtesy turn".
Good luck! Let us know how it goes, and what did or didn't work well!
Cheers,
Maia