Can I take my case to civil court if my attorney assigned to my slip and fall case decided to drop the case after one year?

I don't know what you think you mean by an attorney "assigned" to your case. For a slip and fall case, a client hires an attorney. It is the client who decides whether to drop the case. If your lawyer determined that your case was not worth pursuing, there probably was a reason. If the statute of limitations has not run, you can represent yourself, but I doubt that a person in your position would be successful.