For me, the conversation feels stilted because there isn’t enough of what is going on around them to make me feel like I’m in the room with them. Set up part of the scene with the conversation and see if it helps.
Also, internal dialogue is a speed bump, so only use it when it’s vital for showing hidden thoughts or feelings that no other character should be able to know at that point.
Liz ran her fingers through her damp hair. “Besides, she’s going out with Bill Thornbourgh. You know – Mom’s broker.”
I finished drying off with the towel and threw it in the poolside basket. “Did they go to the Cayman Islands together?”
“He owns a condo down there. Try competing with that.”
I made a face and Liz laughed, flicking her towel at my legs. “Oh, you thought you could compete with him,” she said, looking me over in mock appraisal. “Well, you are wealthy and breezing up the corporate ladder. That’s in your favor. But I don’t know. She seems to go for those bland guys who don’t compete for the limelight. She’s definitely into the whole lifestyle, though – always going on about his place in Grosse Pointe and his Ferrari.”
I ducked at she threw her towel in my direction and headed for the house.
This is just my perception and may not fit at all into your writing style. You’re going to get as many suggestions that vary as you do posts, so with any luck one of them will be what you were needing.
Have fun – sounds like an interesting delimma you’ve set up.