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You can spray the plant with Neem Oil.
Use it weekly until the problem clears up. Neem Oil is safe for people and pets.

Here is a link with more information.
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

Most types of begonias will probably do fine with this plan. But since you won't be there to enjoy them, an easier option for both your begonias and dahlias is to let them go dormant for the winter. If you know the type of begonia you have, look up its winter dormancy preferences.

To let them go dormant, you can either dig up the tubers/rhizomes after the foliage has faded and store them in a dark, dry location, or bring the pot indoors and place it in a dark, dry, cool but not freezing location. Stop watering them to allow dormancy.

You have a Wax Begonia. Many people grow these as an outdoor annual plant, but you can winter your plant indoors.
Dead head the spent blossoms and follow the care information in the link below.
https://www.gardeningknowhow.com/ornamental/flowers/begonia/annual-begonia-care.htm

As this article explains, it is difficult to divide begonia tubers successfully, and a more reliable method is to make cuttings when the plant is growing:

http://www.extension.umn.edu/garden/yard-garden/flowers/tuberous-begonias/

It is possible but can be difficult to divide the tubers of tuberous begonias:

http://www.extension.umn.edu/garden/yard-garden/flowers/tuberous-begonias/

A more reliable way is to propagate begonias from cuttings:

https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonia-propagation.htm

If you have a greenhouse you can start your tubers anytime. It can take up to 3 months for your tuber to flower.
Many gardeners start their plants 1 month before the last frost date of your growing zone.
This link will help you determine this date and your growing zone.
https://www.gardeningknowhow.com/planting-zones/
https://www.gardeningknowhow.com/ornamental/flowers/begonia/growing-begonias-from-corms.htm