I am not interested in surgery, would filler reduce the appearance of deep grooves on the corners of the mouth? (photos)

You did not specify what problem you had with Juvederm, but whatever it was, it could have related to being treated by inexperienced injector, the wrong choice of filler for the area being treated, the wrong area treated to get the job done, etc.
From the photos provided, there appears to be significant volume Loss in the cheeks, which is responsible for not only the flattening of the cheeks themselves, but for the depth of the smile lines and to some extent the deep Marionette lines at the angles of the mouth and even the jowls along the jawline.
So, the use of a robust HA volumizing filler in the cheeks would not only create a more feminine, oval, heart-shaped face and ogee curve, but would secondarily improve the under-eyes, the smile lines, the jawline and the marionette grooves on the sides of the mouth.
Additional robust filler can then be used to further more directly elevate and smooth the marionette grooves still further, which, when the cheeks are now appropriately revolumized will respond much better to treatment. (Injecting the grooves first, without addressing the cheeks first, would be an exercise in futility, as would using simple fillers, such as ordinary Restylane or simple Juvederm; (appropriately chosen more robust fillers are required here). When performed properly, one treatment session of about 30 minutes should be sufficient to yield immediate, on-the-spot results, and engender little to no significant post-treatment downtime.
The bottom line is that you would be wise to consult with an experienced, board certified cosmetic dermatologist filler expert. This is no place for novice, amateur, inexperienced injectors accustomed to just using plain Juvederm or simple Restylane for filling smile lines. Hope this helps and best of luck.

In reviewing your photos provided, I do think you can help this area without surgery. It would take a comprehensive approach which would include not only fillers in the marionette lines (corner of mouth but also the line that goes towards the jawline) but also you will need to help lift the corners by also addressing the sunken cheeks (which will help to lift the corners) and also some product (ie Sculptra) to help lift the lower face which will also help to lift the marionette lines. If you try only to fill the corners, you probably wouldn't see much improvement plus it is possible to be overfilled here which is probably why you had a bad experience. If you are able to do an in person consultation, I could show you more easily the effects of lifting the cheeks and lower face and how that would help the marionette lines. Good luck.