Our older cat's behavior radically changed after the birth of our twins. What should we do?

Three years ago, we had twins. Since then, one of our cats has completely lost it (we've caught her on video so we know which one it is). Frequent urination all over the house, refusal to poop in the litter box (she'll do it right outside or sometimes in another part of the house altogether), etc. If she can get into the kids' room or anywhere near one of their blankets, she will make an extra effort to empty a full bladder. We keep our bed covered in plastic and can't leave any cloth items on the floor anywhere in the house (including bath mats). This cat has peed on expensive furniture, including a glider and a sofa. We've tried behavior modification, but with two toddlers in the house our ability to deal with this problem is limited.This cat was a beloved, constant companion for 13 years, but she is now 16 and has been trouble for the last three years. Should we put her to sleep or let her live out her years? No one would adopt her -- she is old, arthritic and pees everywhere!