Cleaning u-joints after re-greasing

I put new grease in my front & rear drive shafts u-joints every oil change. I've been doing this for years and have always just pushed and pulled paper towels through and around the joint to get the squish out. It was about 90°f at 8 am this morning in SC and I'm laying there under my wife's 4Runner with weak hands due to being on chemotherapy for the last few months, trying to clean up the old grease that squished out and I couldn't help but think that someone has to have a better way to do this.

I thought I might use an old toothbrush to push shop towel or paper towel through and around better, or a popsicle stick or maybe use a bit of cording with a loop to pull shop towels through the tight spots. I'm normally squished up under my Tundra or her 4Runner, laying on cardboard with less than an inch of room between the stainless Toyota side steps and my chest, I (6'3", 185lb, athletic build) wiggle under the vehicle and then have maybe between 2 and 4 inches of room to work while I'm getting burned by the exhaust pipe I forgot was hot. It's simple and works better than me trying to get each vehicle on jack stands, which after the cancer treatments I just don't have the energy for.

Anyone have a better method that they use to get the squished out grease off??