If you botted on a different IP to the one you usually play on (home, school, work etc.) you could claim your account was hacked. I know some people have gone all out suicide madness botting hundreds of hours straight on their main but on an IP located on the other side of the world. Once they're banned they just say 'oh I had no idea, I was hacked and now I come back to play RS and I'm banned for botting'. That works a lot of the time, but needs a little forward thinking. So not sure if that's something you can do in your situation.
I've never had any experience with the 'I'm sorry' route. Even though I've got one 2-letter name account which was banned for botting (lol don't ask) I never got myself to appeal it, I'm sort of doubtful it'd work but I figured if I only have one shot at it I'd make it count so I've been waiting a couple years now in hopes it'd increase my chances of success.
In any case, the whole process takes time and you usually get a reply after about a month. If you're going to wait before appealing that only adds whatever time you wait to it. I'd say, make a new account and play on it in the meantime, if you do get unbanned then that's great and you'd have two accounts. If not, then at least you didn't waste any time before starting over.