Yeah, if you could get those grades up you should have no problems. Freshman year grade troubles are easy enough to write off if you improve on them and do well in the later years. And if you’re worried that your GPA is still low by the end because the 2.8 brings you down, you can (and probably should for ChemE anyways) go for a Masters and get a higher GPA there.
Community college for 1-2 years is a decent option for saving money and still getting a very solid lower division education. Most universities are pretty good about taking transfers who did well in community college. You could also see if you could improve within UW. Would take another year, if you’re even allowed to take that option, but that’s not the worst thing that could happen.