So, there's really isn't any way to tell what is in the cup, but the babysitter said it was the medicine.
I'm thinking that TS and EJ did have a plan, for whatever reason it involved San Antonio.
The plan was to go there and wait for LM to sign over his rights. Perhaps TS knows of someone there to help with illegal documents, (maybe that's why she needed a sitter...she couldn't bring a baby to meet this person(s), it would raise too many red flags...or she had to pay the man with a "favor"). That didn't happen, obviously, so EJ, being desperate, has to do some looking online.
How in the world are you going to do that much research with a baby? Make him sleepy. Drug him up with Tylenol and Benadryl while you do your research at the libreary (can't do that with a fussing baby) and lobby, making calls on a prepaid phone you got.
You find someone (either on your own or with the help of TS, or even someone you considered before TS), but they can't get there right away, they may not live there. The plan is set, but you have to cool your heels for a few days. You are going to meet the couple at a park. So you snap a few more pictures before he's gone from you forever.
The new couple, excited to have a baby at any cost (jailtime included), bought their own carseat (or it is one they had already from their deceased/failed previous adoption). This couple could be an older couple with no family and a home with distance from neighbors, so it won't be hard to hide him for a while.
EJ knows how to make it without money, finding shelters and such, from her life in foster, so she doesn't have to worry about that. She agrees to a small amount of cash, to get her to her destination. She was desperate, so it's not like she could demand a ton of cash. So the couple gives her a few hundered. EJ leaves her car and is hoping, her previous life, behind as she hops a bus for sunny Florida. From there, she can go anywhere she wants, or just hide out there and get a job.
She's had a lot of people in her life let her down, and the feeling of relief comes over her as she arrives at the shelter in FL, so she logs in, and updates her status for all those haters and describes her mood as adventerous, because that may actually be truthful.
Just thinking out loud.