I studied to be a Machine Design Engineer, then switched to Manufacturing Engineering for a MS because I wasn’t interested in developing the super strong math skills required for the dynamics of nitty gritty manufacturing machine design. I got into electro-mechanical product design through an internship, and just kept on down that track.
In my opinion, you DON’T need an MS, don’t need to go back to college. You need to spend time in a design software, designing, building a portfolio and competence. Paper, Pencil, Calipers, a ruler, and something to practice modelling is all you need. Measure the features on the part, and recreate them in 3D. Find discarded products to take apart, I used to collect broken electronics and products (ink-jet printers for example) and dismantle them. Good design is all around you, study it.
The key to success in design, is not a prestigious degree, or internship. It is generating as many ideas as you can to solve a problem, and quickly figuring out which ones are going to be trouble. Then, advance the plausible solutions in parallel until the solution reveals itself.
Start where you are at with what you have to solve a problem. And keep doing it. Design will always be using what you have immediately around you, creatively, to make something that wasn’t there before.
No one needs to crown you a designer. Just start designing.