The first thing to do is to talk to your academic advisor and the department chair, in that order, and see if they can either 1) waive one of those course requirements for your major and substitute something else, or 2) change the schedule for one of the courses so they don't conflict. Even if you're not supposed to take them in the same year, there are people who need to retake courses or get out of sequence, and unless one is a prerequisite for the other, it just doesn't make sense for them to have ever been scheduled in the same time slot in the first place if they are both required for the same major. If they change the scheduled time, you may not be able to sign up for it until after the schedule is changed, but can probably get something from the dept chair giving you permission to register for it late if you sign up for a filler course in the meantime (one you'll drop once you register for the required course, and is only on your schedule to satisfy your credit load requirements for full time study).
Another option is to find out one of the courses, or a similar one that will count, is offered during the summer, either at your school, or at another from which you can transfer the credits for your major.
If none of that will work, you can try talking to your dean for academic affairs (or the equivalent, whatever his/her title). They may be able to work with the scheduling office to change the time of the course if the dept chair can't.
The absolute worst-case scenario is that you're stuck having to delay graduation for that one course. If that's the case, then if you're receiving financial aid of any kind, make sure you discuss this with your financial aid office now so you can plan ahead for how that will impact that financial aid...will it cover a 5th year, do you need to remain a full-time student, will any of your loans start coming due for payment if you drop to part-time status in a 5th year, etc.
What do you plan to do after you graduate? If you want to get a job, you could use that extra year to get an internship with a prospective employer so that when you complete that last course, they are more likely to hire you on full-time. If it's graduate or professional school you want to attend, then explore research opportunities. It's great to have a student come along who only has one class and can therefore dedicate the rest of their time to working in a lab. Basically, that year doesn't have to be wasted, just start looking for opportunities that will enhance your qualifications for the next step after graduation.