My cat stopped using the litter box. What to do?

My cat just turned 14 years old. About 6 months ago she stopped using her litter boxes (we have two different ones). This caught us off guard because we have had the litter box in the same place for practically her entire life. It is in the garage, and she can access it by going through a doggy door. Our first thought was that the litter box was too difficult for her to reach now that she is older (this ended up not being true. She still knows how to use it, but sometimes refuses to). We moved it in the family room to a quiet corner that is easier to reach, and she still was not using it. She would often poop right in front of the litter box. We took her to the vet thinking it was something medical, but she came out with a clean bill of health (except that she has hyperthyroidism, which both the vet and I have known about for the last year and a half). We took all the advice the vet gave us and tried them out. We took the hood off of the boxes, we adjusted the amount of litter in the boxes, we used a special litter that is supposed to attract cats, we got diffusors/sprays that are supposed to relax the cat, we got a third box...nothing has worked. We even tried confining her to a room with the litter boxes at night and she still would not use them. 

One thing I have noticed is that she likes to go to the bathroom outside. She will (sometimes) sit in front of the door and meow for me to let her out so she can do her business. Sometimes she wont even do that and just poops or pees right in front of me without warning. I have thought about filling a box with soil and using that as a litter box, but then the smell would travel all around the house and it would be difficult to clean since it doesnt clump. 

I dont know what to do at this point. I definitely do not want to put her to sleep over something like this because her quality of life is still good. But I spend at least one or two hours a day cleaning cat urine and feces from furniture and carpets. Does anyone have any advice, or has gone through something similar? Any help would be appreciated!

Her personality hasn't changed, she isn't experiencing confusion, sleeping less?  

No. She is acting the same she has been for years. 

That really hard to cope with, I had cats do that kind of thing mostly pee and I would say that despite the clean bill of health that their health went downhill from there rapidly. In one case a 10 year old male over a one and a bit years kept doing it but we actually weren't sure it was all him, and the other was a 26 year old female was checked a few times even given pain relief would just pee on things, but with in a few months she just lay on the floor crying a sad meow like she could not get up , we took her in that day / hour really and that was the end.
I can only suggest boxes or baking trays that have really low sides amd no litter right by the front door maybe. Yes there is no help with the smell but better than the carpets!