For towing trailers and carts, the 124 will do just as well as your 1810. If you haven't physically looked at the 124, be aware the steering and clutch will most likely need work. This is more due to neglect and 40+ years of wear, not inferior design.
As far as the 1810, there are a few options that should make finding a replacement easier. You can repower with a vanguard or magnum or ch kohler out of a cub. I know of at least one member here that even has a k series kohler in a 10/11 series tractor.
The right answer is to buy the 124 and find a new engine for the 1810 and keep them both.
Welcome to the sickness.
Bill