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Hm... I would almost say that this is fungal. Make sure that the soil dries out slightly in between watering. Apply wettable sulfur to the soil, along with a little dolomitic lime.

Here is an article with more information on the care of this plant: https://www.gardeningknowhow.com/houseplants/flowering-maple/growing-flowering-maple.htm

It is best to avoid pruning until later in spring/summer and to wait and see if the plant produces new growth. Many cold-damaged plants will recover on their own. After it puts out new growth, you can see if there are any dead areas and prune them out.

You might also want to check whether the drainage around the Abutilon can be improved, since flooding can also damage plants. Here is more information on soil drainage:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/yard-drainage-problems.htm
http://goodlifegarden.ucdavis.edu/blog/2011/08/how-to-improve-your-soil-drainage/
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/making-sure-soil-drains-well.htm

It looks like you have a pest issue, likely a scale insect. The black on the leaves appears to be Sooty Mold, which is the residue from pests.

Treat the plant with Neem Oil.

Here is a link with more information.

https://www.gardeningknowhow.com/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

I like using alcohol on swabs to remove as many as possible.
Then treat plant with Neem Oil.
https://www.gardeningknowhow.com/plant-problems/pests/insects/control-plant-scale.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

Mail order is possible if not available at your local garden center. I searched "flowering maple for sale" and saw several, ebay and other sources:

http://tom-piergrossi.squarespace.com/abutilons-flowering-maples/

https://www.gardeningknowhow.com/houseplants/flowering-maple/growing-flowering-maple.htm

Unfortunately, this can have many causes. Can you please include a photo? This will help me to see what is going on. Have you seen any insects? This is a very common cause for sap production. Another cause can be overwatering. You should always wait until the top is almost dry to water again. Make sure the container has good drainage, or it can lead to root rot.