Where did you get the idea that there is no competition? Even though it may appear that you are the only advertiser, that may not be true. You left out a lot of details that would be needed to give you an accurate answer.
Are you using broad match keywords? That could be an issue that causes your QS to drop.
Have you produced a Search Terms Report and verified all of the keywords that trigger ads? It is probably an issue of targeting keywords that you did not intend to target.
It could be other things related to your overall account structure, or how you have organized ad groups, or which network(s) you are targeting, or your ad text, or landing page issues. We need specifics from you to get you a definitive answer.
Google has indeed made significant changes to the AdWords system that require changes on your part to maintain performance if you just set and forget it going to cost you a lot.
Your keywords’ Quality Scores are based, in part, on your overall account Quality Score. If any of those other campaigns have lower QS, and it is likely they do, those other campaigns may be dragging down the overall account QS which in turn lowers your QS for your specific keyword that has no competition.
Another thing to consider is that Google changed significantly the ad triggering algorithm a while back, so while you are using exact keywords, they are still likely to trigger ads for keywords that are not an exact match. Sadly, the change that AdWords made a while back tends to trigger related keywords that are not always relevant. You can confirm this by running a Search Terms report for the past 90 days.
To correct the issue of triggering ads for irrelevant (or less than extremely relevant) keywords you can go into your campaign settings and disable the option to “Include plurals, misspellings, and other close variants” under the “Keyword matching options”, section.