Baron 58 pilot reported during taxi they stopped to clean the windshield due to poor visibility. The aircraft continued to roll off the side of the runway into the mud.

Narrative

I was assigned a series of flights by Dispatch in aircraft. The first flight I flew it empty from ZZZ to ZZZ1 to pick up cargo. It was flown under Part 91. The weather in ZZZ was categorized as LIFR; ceiling of 200 FT and visibility of half a mile. I picked up ATIS; taxied to the exit section of the ramp and called ZZZ Ground. I received my IFR clearance and taxi instructions. I was instructed to taxi on '1' and hold short of Runway XX at '2'. I complied. Whilke taxiing on '1'; the windshield became misted over and visibility was labored. The windshield needed to be wiped. I opened the pilot storm window while slowing the plane with light brake pressure. My intention was to stop the airplane gracefully without skidding. As the airplane slowed down farther; I thought it had stopped as I reached out to wipe the windshield. It then became evident that the plane was off the taxiway to the left. Due to the wet ground; the left main gear and nose wheel had sunk into the mud significantly. Only towing equipment was going to be able to move it. A tug vehicle was summoned and moved the aircraft back to the ramp. Upon returning; the mud was hosed off the tires; inspected and returned to service. Another pilot continued the flights. Taxiway 1 is significantly narrower than the others. The lesson to be learned is that patience and being completely aware of the airplane movement is essential. The airplane was not completely stopped as I believed it to be.