Two things I would immediately start researching - impulse control training, and leash aggression and how to avoid developing it.
Impulse control - these are fun exercises and segues well with NILIF (Nothing In Life Is Free) or Learn to Earn lifestyle.
The Learn to Earn Program: Developing Leadership in Humans and Impulse Control in Dogs | Animal Behavior and Medicine Blog | Dr. Sophia Yin, DVM, MS
This is a great site for some games to make it fun
Impulse Control Training and Games for Dogs | ASPCA
As for the play issue. I would not try to do this alone with you and two dogs.
Re-directing a reactive dog, takes a huge amount of time - and if the reactivity develops into aggression because every time she's on a leash and another dog comes around, her play drive is frustrated ... and she is corrected... well... lets just say, you could create more trouble than you already have.
On-Leash Aggression, Leash Reactivity, Dog Aggression
With a reactive dog, the key is to discover her boundaries - where/when does she become reactive. How close does the other dog have to be before she looses her focus on you and turns to focus on the other dog. Day after day, training session after training session, you want to put her just shy of her limit of reactivity. You want to prevent that reaction from EVER happening. But if it does, that's OK - just take a step back to where she doesn't react and start again.
Always end on a good note even if that means having to go to the other side of the house where no dogs are in site.
I didn't have a reactive dog. I just had a rude one, with no recall. He was wonderful when he was on a leash. That really helped me. Made my work much easier.
The way it was explained to me was, you pick three levels of behavior.
One - good behavior, everything is fine.
Two - behavior that is rude but deserves a second chance and correcting or redirecting ...
Three - the absolute no-no. Play ends now.
Be sure to keep these levels and behaviors that fall into them, very clear in your mind.
Any behavior that falls into level three - you mark with a phrase or word - I use "Too bad." And you immediately remove the dog from the area. To a "time out". The time out is not quite the same as with a human child of course. Because a dog isn't going to think about what it did ... it's just going to link behavior #3 with being taken away and left somewhere that's no fun. Soon they learn that's not such a fun behavior - just like they learn that good behaviors get treats!
If your dog does something within level #2 - you call them off wait a couple of minutes and let them go back to play again once they've settled down. Maybe after performing a couple of things that they do know (ie: sit/treat, down/treat).
Now I know, it's hard to pull a dog out of the play when they are running full tilt across the park. So you first have to keep her on a leash and develop a strong recall. You don't start with anything until you have the basics.
For you, it will be the same, but you won't get as far as dog on dog play until you eliminate the reactivity.
Personally, I think it will be better for you to do this with a trainer who has a great deal of experience working with reactive dogs in a positive way. If this "free" trainer has known this dog all it's life - that doesn't bode well, if all he said was to keep the dogs separate. He should have explained a little more history than that! Haven't they worked on her manners yet?
You want to find a trainer that is going to help you teach her what TO do, in place of what NOT to do.
ie: you want to avoid a fail. Do things that will allow her receive praise. If you push so hard that she fails and does the wrong thing, you've pushed too far. Lead her away from it with no other reaction and give yourself the correction.

That's a joke. But not really.
I do believe in corrections BUT only after the dog knows what you want. Right now, your girl has no clue what is expected of her. So you're starting from scratch. You have to develop a bond where she trusts that you have her best interests at heart. That you want her to succeed, and you are going to show her the path to that success, not just punish her for doing the wrong thing.
See what I mean?
Is this free trainer a member of APDT?
I might start here to find a good trainer in your area.
Association of Pet Dog Trainers - Dog Training Resources