While I believe in helping when and where we can, I also believe in being honest---with myself and others.
Here's what bothers me about this (and I realize I only have part of the story): It seems like she EXPECTS you to watch him. The comment about coming to pick up his meds after you told her you would get back with her seems manipulative to me. Maybe it isn't. Maybe that's just me reading into it, but my radar always goes off when people make presumptuous statements, and her saying that sounds an awful lot like she assumes that you're going to agree to do this.
The next thing is her list of DOs and DON'Ts when it comes to watching her child. If she trusts you enough to ask you to watch him, she should trust that you know how to interact with him. I mean, you're relatives; you've been around him. Does she expect you to handle him with kid gloves and allow him to tantrum and break things if he doesn't get his way? I'd sure want to clarify that part before agreeing to watch anyone who needs this level of care.
Next--- did she just find out about the trip? Did the ex just ask her today? I'm wondering
why she didn't make arrangements or ask you well ahead of time. Did other arrangements fall through? If she didn't have any other arrangements and she knew about this trip ahead of time, then to me, it seems like she just dropped this request in your lap at the last minute and expects you to say "yes." If this is the case, and if you do say yes, then you are setting yourself up to be her last minute babysitter the next time she needs one.
Maybe I'm wrong. Maybe the ex just let her know about this trip today. Maybe she does really trust that you can care for her son, and that's why she came to you. So, even giving her the benefit of the doubt, the next issue that comes to mind is the one of being true to yourself.
If you truly believe you are not up to this task, then you must be honest with her.
The title of your post, "I want to say NO," and saying, "It scares me to keep this other young man" and feeling like you don't have the patience, even with your own son---all these things point to you not being up for this at this time. And for those reasons, you must be honest with her.
All you need to say is that you are not up for this level of caregiving at this time. You are drained from dealing with your own son and don't believe you can give her son the additional care and attention he requires.
It is good to help others, but sometimes, we cannot. That's okay. What would not be okay would to be dishonest and let her assume you're up for a task, which by your own words, you are not.
Help her to find other care. Maybe you could arrange for a lunch date with the caregiver and her son while your cousin's away. That way she knows he'll have a little contact with family, but you won't have to be the caregiver for the weekend.