I don't understand why you think you'll fall behind. You are required to take atleast a couple electives before you graduate, and it will probably be beneficial to you to take some easier classes now while you learn some study skills and get settled in at your school. It sounds to me like you breezed through high school and are realizing the same strategy isn't going to work at a university. And to be brutally honest, pre-calculus is not a difficult class compared to the 3000 and 4000 level classes you'll be taking later on.
I strongly suggest you take time now to explore your options.
As for psychology/pschyiatry, there is a big distinction. Psychiatrists take 4 years medical school and must complete a 4 year residency. They are licensed medical doctors and can prescribe medication. They also make a lot more money than psychologists. If this sounds like what you want to do, I would stick with BioChemistry. It is a much more reputable degree than psychology and will do a lot more to prepare you for medical school.
On the other hand, and I'm assuming you're having visions of chilling with a clipboard and listening to people whine about their problems, there is clinical psychology. You would major in Psychology (bad news in my opinion, if for whatever reason you don't go on to graduate school a psych major by itself is pretty worthless) and then need to get a PhD or PsyD.