I did the same thing, switching to using the other hand and arm for most things. It did not take long to master using it and has helped tremendously. Also adjusted the seating position at my desk to help with elevation and move the mouse closer so the shoulder is supported (adjusted arm rests too), so as not to strain either helped with other tasks such as writing (better support for dominant/bad arm-shoulder) and making using the mouse are easier as well before switched. If your work station is not ergonomically correct for your seating and desk height, can you ask about getting help with it or evaluated, same with the kind of mouse you use? Also moving things closer and within better reach helped, especially when picking things up so the arm stayed closer to my body helped to prevent the straining. However using the other arm helped the most, giving the other shoulder a needed break. Hope this episode passes soon!

PS I am right handed and my mind could not adapt to using a left handed mouse. Most mice are both handed now but in case you are using a right handed mouse now, it was easier to use a right handed mouse in my left hand, than adapt to a left handed one (in case you do get evaluated and they ask about switching you) You might find this easier as well if right handed. Good luck!!!