Right rear turn signal will not work.

simple open the trunk lid there is a plastic panel right in front of you carefully remove the clips that hold it on this will give you access to the bulb socket turn it to the right and out it comes the bulb pulls straight out, wear gloves to handle the bulb as oil of your hands will cause hot spots and shorten the life of it , put a dab of dielectric grease on the back of the bulb and repeat the process to install

Your right rear turn signal bulb is blown.Replace the bulb and the fast blinking will stop.

What that rapid turn signal on the dash means is, the rear turn signal tail light is burnt out. This is how you know the bulb is bad. Replace the burnt out tail light bulb and you will be back to normal.

One or both of your turn signal bulbs is blown.

You need to squeeze the little 'fingers' on that white piece to allow the taillamp assembly to be removed. See the YouTube video at the link.
How to remove tail lights in Dodge Avenger 2011 up

Just because the bulbs look fine, doesn't mean they are. Start with the bulb replacement. If this doesn't solve it, and it has under 36K, it should still be under factory warranty. Shouldn't cost anywhere near $200 either way.

make sure you have the correct replacement bulb for that car. Some bulbs do more than one thing.

flasher unit, available at parts stores cheap

replace the turn signal switch it self, mine did the same thing with the left turn signal, put in fuse and it will continue to flash... went to junk yard pulled one out of a 97 avenger just like mine, swapped them out and now it works fine. looked to be like some corrosion... to repace: 3 screws under columb, 2 screws on side of signal switch (black plastic side). unplug and replace

try to replace the flasher. it is round and has either two or three prongs on it. it should be by the fuse box.