My understanding is that, although possible, if the cancer risk does exist, it's very small; here's an article placing it at 3-4 cases per 1,000,000:
http://aaem.pl/fulltxt.php?ICID=1011054
(They cite a 2012 meta-analysis which showed no statistically significant effect of mouthwash)
Here's a recent large-sample statistical study:
https://healthcare.utah.edu/huntsmancancerinstitute/research...
They find an 1.15 odds ratio of cancer for mouthwash users, but in the non-smoking non-alcohol-drinking cohort this effect goes the opposite direction (OR 0.83); statistics is hard and there isn't high-quality data to give a real conclusion. But since the overall incidence of oral cancer is roughly 1 in 10000 (
http://www.nidcr.nih.gov/DataStatistics/FindDataByTopic/Oral...) I wouldn't be too worried; all these show is that the effect is tiny. (Of course, if you only remember the big headline from ten years ago with a tiny sample size...)
There's a clear line of research showing benefits in mouthwash:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4948552/
It's true that the fluoride in mouthwash does little, and there are some stupid mouthwashes that don't have any real ingredients, but essential oil mouthwashes (like Listerine) and several others (as mentioned in the review) have been shown to have significant gum benefits.
There's also some interesting research with tetrapotassium pyrophosphate and tartar buildup: http://periogen.com/periogen-studies.html Although I expect these studies are manufacturer-funded and thus overstate the effect, it seems pretty hard to deny that it helps.
Note: I'm not a dentist either, but I read this book my dentist recommended: https://amzn.com/B009LPLMRU. It's well-referenced for everything except water fluoridation and her conflict of interest (xylitol products).