How do you deal with an 'on the way out' pet?

I have never found the strength to have any of my furry family members put down. I have managed to sit with them as they breathed their last. In March of this year, Sparrow, my 14 year old cat was dying of a brain tumor. My wife and I visited our daughter, son-in-law, and grandkids in Indiana for a week. When we returned, Sparrow had disappeared while we were gone. Our roommate told us that two days after we left she ate her breakfast and went outside as usual and didn’t show up for dinner. I wish that we could have found her so that I could have buried her in the back yard. I miss her friendship as she helped me through a difficult time in my life.
We now have taken in a stray cat who showed up at our front door and cried for three days straight before I finally gave in and fed it. She follows me around outside and helps me do all of my projects.