You have to look at the individual school, vis-a-vis switching into engineering. A few allow you to switch at will; many make it quite difficult, and at some it’s nigh-on-impossible. As you might suspect, the schools with the greatest flexibility are often more wealthy and competitive institutions, as inflexibility is more cost-efficient for colleges!
Is your 3.6 GPA weighted or unweighted?
Since you have funding, you may want to look at more expensive private U’s where being URM will give you more of an advantage. (Most public U’s aren’t allowed to consider race in admissions.) Think about applying to Lehigh’s Diversity Achievers fly-in program when the application opens https://www1.lehigh.edu/admissions/undergrad/visit/dap They’re working to recruit more URM students - might be a great school to give you the start on engineering that you’re looking for. (Plus it’s also strong in liberal arts, business, and art/architecture/design so there would be many options if you should change your mind.) Clarkson and Union in Upstate NY would be great places to apply. Drexel and RIT both have terrific co-op programs. WPI and Rose-Hulman are both smaller schools with project-based approaches.
More diversity fly-in programs that you might want to apply to:
Bucknell (LAC+engineering) https://www.bucknell.edu/journeytobucknell
Union https://www.union.edu/admissions/visit/getting-to-know-u/
Trinity (CT) (LAC+engineering) http://www.trincoll.edu/Admissions/campusvisit/pages/preview.aspx
WPI https://www.wpi.edu/c/discover-diversity
Miami of OH (so-called “Public Ivy” with good engineering) http://miamioh.edu/admission/high-school/bridges/
As for transferring to a better school - some schools take a lot more transfers than others. State flagships often plan for a lot of incoming transfers because of community colleges feeding into them, and some private U’s take a relatively larger number of transfers (Northeastern, USC, Cornell). Other elite schools take only a trickle of transfers, because they don’t really expand the class over the four years, and not very many withdraw. With engineering, there’s also the issue of getting programs to align so that you don’t end up having to repeat required courses.
Another route you could take is an established 3:2 engineering program - 3 years at a liberal arts college with a transfer path to a top university to complete the engineering degree in another two years. The plus side is that this would give you more time to ramp up in math and science, as well as time to explore other interests, before transferring, and many programs guarantee transfer admission if you meet GPA guidelines. The downside is that those GPA requirements can be quite high (3.5+ in pre-engineering classes in some programs), and a majority of aspiring 3-2 students don’t end up transferring, either because they change their minds about engineering, or don’t meet the GPA requirement, or don’t want to leave a year before their friends graduate when they could just finish their BA/BS where they are. But, it’s an option that could give you the ramp-up time you need and give you an established path into a university like Columbia, WashU, Dartmouth, etc. Here’s one example from U of Richmond, which also has a 3-1 program with UVA, in addition to it’s 3-2 and 4-2 programs https://preengineering.richmond.edu/dual-degree/index.html