I think you've gotten some good thoughts. Erections are normal. If you really don't want one and ya, it could embarrass both of you, I get that, then don't play in those ways. (whatever ways they are). Or wear loose pants so it's not noticeable and step away for a second.
Just be thankful your penis works as designed and enjoy it. Your girlfriend should be flattered. :) There may come a day many years from now where you'll wish you could get erect. (I hope not, however!)
You don't mention your age, are you a teenager? Also, maybe, clarify what kind of play you are talking about? -- Tickle? Hug? Joke around? Play a game? (Or something more overtly sensual?) It's pretty natural for someone to get erect when he's with his girlfriend in certain situations.