Given where you live - try some in the shady garden and see how they do there. And you don't have to lift them, either. Most of them will come back again next season. I'd say the peat in the mix would be the problem. It can be very hard to rewet once it's reached a certain level of dryness. You could try adding a soil saturation mix; or some sterile worm castings which act like a fine water-holding soil; or those water crystal thingies; or add some coarse horticultural pumice about 3mm-5mm sizing, not too many fines, to your mix. Personally, I'd go for the worm casts and the pumice for the aeration, drainage, and water-holding properties. You might also want to review your basket lining construction to see if another system would give you better results. I used a thin layer of foam rubber last season, inside a well-holed plastic cover, and it seemed to work. I've also seen a posting by someone who used those 'overnight baby diapers' - with the capacity to hold moisture and slowly release it - as a discreet option in a hanging basket. Carefully smothered by coir or sphagnum - it could work. I don't know anyone who uses them at the moment so I couldn't try it.

I would say we would need more information than a picture of two old (dahlia) leaves. How wet are your pots? How much sun do they get? How dry do you let your pots get? What kind of begonias are they? When did you get the begonias (this year?) Did they grow at all or slowly die down? Did you remove old dead leaves as they fell or turned yellow? A picture might help too.

Oyama pots wick - you might need to lighten up your soil with perlite, as you can't wick with standard nursery or av soil - it'll remain too damp due to too much peat. i put my begonias on wicks and they do fine. however, i use AV soil and add about 30% by volume of reg. perlite - this is pretty much standard practice for wicked AV's. the recommendation is at least 40% of perlite for aeration to prevent dampening. i do allow the soil to dry up on the surface a bit now and then - perhaps once a month. so, when the water reservoir runs dry, i don't water . and then do a light re-wet from the top and start the cycle again.

Kyle has a good idea. I typically put a large rock either on the rhizome or close to it. This helps in two ways - 1) puts an anchor on the plant until it roots and can be on its own and 2) discourages squirrels from digging in fresh soil. This photo is not a begonia and is not to weight down a plant for rooting but is showing caladiums coming up around pottery shards (for discouraging digging rodents) These are exceptionally big rocks to hold some broken off stems of popenoei a few years ago more broken shards holding a big fat rhizome of something in place. rocks in a pot of carolineifolia I am trying to do a bonsai pot with a broken stem of Selph's Mahogany indoors. The stem was leafless for weeks and the broken end rotted off. I sparingly watered it and put a moat of water around it and covered with a dome. Once it rooted and leaves came on, it started to flop over so I now have the rhizome propped up with some large decorative stones. one final example of this big Immense plant held in place with a good size rock in a pan of perlite.