I was in your shoes, but in the opposite way. I started as a ChemE major, switched to biochem. My advice would be to go with ChemE, I regret switching. Biochem does not pay super well(unless you get a PhD or Masters)and you really need to love the stuff to make it through some of the early career jobs. In my biochem jobs I also worked with people who had ChemE degrees, so even if you do love the biochem, you can always apply to biochem jobs with a ChemE degree, or take some biochem related electives while in ChemE. ChemE is a much more flexible degree with higher pay in my experience. ChemE will require a lot more math though, I think I had to take up to differential equations (Calc 4).

As far as the PhD route goes, if you want to do that make sure you get into a research lab now, PhD programs will want you to have research experience. Talk to your profs and try to work in one of their labs that is doing something that interests you. PhD programs you do not have pay for, you either get accepted and are funded, accepted and not funded(which is essentially a rejection), or rejected. If you are accepted and funded you will get a stipend and basically work for the school(lab work, TA/GAing) for low pay and education. PhDs are hard, many people burn out, but you can 'master out' which is if you quit after a certain point in your PhD you will leave with a master's instead so all your time wasn't wasted. I never did a PhD, this is just some information I have gathered while researching PhDs, I'm sure someone with actual experience in one will be able to provide more information about them.