So you fell down, and a fracture was ruled out without an X-Ray. That's possible I guess, just by feeling the hand and looking for swelling. But the doctors thought there could be some tendon/ligament damage and didn't request an MRI, just a ace bandage? Sounds like great doctors to me. 6 weeks is a long time to have pain, go back to a doctor and say its been like this for 6+ weeks even though you've rested it. Rest obviously is not working, so if they tell you to rest it push for an MRI or something that could prove them right or wrong.
I tore a tendon out of its cartilage sheath in my right wrist, waited to see a doctor for a month because I thought it was minor. Saw one and he thought it might be sprained so rest for 3 weeks. Did nothing, saw another doc who though the cartilage might be torn so put me in a splint/brace, didn't work. Saw another who said it was likely torn tfcc, said the doc who was qualified to do surgery was in a different city so I went to him. This guy said the tfcc was torn but wasn't causing my pain, said it was the tendon out of place. So had to get surgery.
Long process, but could've been much quicker if I would've realized months of pain without activity was a serious problem. Also, the wrist has a lot going on in it.
Find a wrist specialist, not a general doc. I could lift through my pain but it was and is a stupid idea. Be aware if it is any cartilage/ligament damage it takes a long time to heal.