When you are placed in Low Priority Que, you are placed there for a set amount of games, its usually 5 i believe. It doesnt matter if in that game you didnt afk or DC. its the games before being placed in that Que that led to the punishemnt. Any Low Priority i ever had were all because of DC issues when my internet was bad, and its been awhile, but I believe it tells you how many games you have left each time. They might give longer for repeat offenders. Nothing you post here can prove whether or not you AFk or DC in a lot of games leading up to being placed in Low Priority. Only Riot has that info.
13 games seems like a lot...why wait that long to seek help? Only thing you can do is put in a support ticket....you might not have to wait that long for help because they even have Live Chat Support Help now at the bottom of where you put in the ticket. if something is wrong they will fix it.