I'm a senior physics major at a large state university. I'm looking towards graduate school but have been looking for some jobs as a plan b just in case. I have several friends who are burned out by academics and are looking for full time jobs right now.
From what I've gathered, physics majors are extremely employable. Back in the day (per-recession from what I gather) jobs were common, so a company diversified by hiring the occasional physics major instead of an engineer. They weren't hiring physics majors in huge numbers, but there weren't huge numbers of physics majors to complain.
The problem is that few people are hiring these days. More to the point, the physics major skill set is not directly employable. Yes, all us physics majors took circuits. We learn about transistors, op-amps, filters and what not. But we can't compete with EE majors who spent most of their degree on that stuff. And there are plenty of EE majors (young and old) flooding the market.
But not all hope is lost. I've talked to a few people from the classes that graduated before me (and who had no luck getting jobs). Most of them have enrolled in engineering masters programs and have found the work much more straightforward compared to their physics degree. I'm not sure what their job prospects are going to be when they finish, but they claim their physics background was much more rigorous then their current engineering classes.
If I were in your shoes, I would try and double. I've taken a few engineering classes and talked to some friends who are doubling in either physics or math with an engineering. None of us think engineering courses are anywhere near as intellectuality demanding as high level physics/math courses (... flame war?). I'm not trying to say engineering is harder/easier. Just more... straightforward. ABET accreditation seems to have standardize exactly what students have to learn, so professors and textbooks teach directly towards that.
Few other thoughts:
1. the physics degree seems designed to prepare you for one thing: physics graduate school. Engineering had to deal with real-world/employers, so they changed their curriculum to insure graduates would be employed. Physics undergrads got funneled by physics professors to work for other physics professors in physics graduate school.
2. if you drop engineering, pick up programming. the physics degree seems to have been built in the 50-60s, so its overwhelmingly pencil-paper.
3. I, personally, would have doubled with computer engineering if I could go back. In part for job prospects, in part because I want to learn more about programming, electronics, and signal processing. These topics would probably help me in my graduate research. (again, too much pencil-paper).