A while back (not so long ago) I too felt the need to add a puppy in my life, and proceeded to purchase an 8 week old Cockapoo who I named Ty. Ty is now ten years old and we are best of friends, though I can honestly say that two events over the years have changed my opinion with regard to finding a pet. First off, a friend of mine passed away 6 years ago after a quick battle with cancer. He left behind a son and daughter (both in their thirties), and a twelve year old cat named Lucy. For whatever reasons, neither of his children could care for Lucy, so I adopted her. Lucy, now over 19, is still going strong. She loves to cuddle (she sleeps above/on my head), and a couple times throughout the day she still shows off her agility and speed by bounding from couch to table, and racing down the halls in an attempt to egg on Ty and Nelly, my more recent adoptee.
If there are ever questions or doubts in your mind as to whether you should give an older pet a second chance at life, I'll be sure to pass them on to Lucy & Nelly, who will both be more than willing to educate you on the pawsitives!