The Clock That Runs Sideways. Clock Chronicles, #2

An exile from a dystopian future is thrown back into the 21st century. A reluctant time traveler, he is determined to avert the catastrophic era that expelled him.  His best chance is a smart, beautiful geneticist.  She may turn out to be an enemy or a lover -- or both. It's a dangerous romance. His only other ally is a time-shifting stranger who is trying to steer him around multiple disastrous possible futures.
The conflict is turning into the exile's personal time war. There are other strangers, crowds of them, some are trying to help him, most are trying to stop him. Dating is  hell when you're a villain, or a hero... depending on the timeline that counts you as part of its history. He manages to recruit  help from  a major corporation by force-evolving a defensive  software system. That same artificial intelligence computer will eventually count as one of his ancestors, since he is partly a product of human design. The initial time anomaly is at an isolated beach on Puget Sound.
The  final anomaly is a confrontation in the middle of the Seattle. The well-meaning, time-shifting stranger is barely ahead of the complications. To say nothing of the dog.