Sorry for taking this long to reply. Lots of errands to do.
An embedded programmer is a catch-all adjective I think. One could be working on very low level programming of microcontrollers (e.g., PIC32, MSP430) and FPGAs [11].
On the other end of the spectrum, you can find embedded programmers working on embedded devices that hosts complete systems with OS, filesystems and utilities such as the ELDK (referred in previous posts). The duties for such programmers include creating device drivers for the specific devices that the product has. I have for example, modified an existing application developed originally for a system with no OS into a Linux application running on a PowerPC board. For example, some of my duties were to modify the original application's absolute FLASH memory references to a method compatible with what the Linux based system offers (hint: I used mmap).
I would guess that someone like yourself with limited exposure to electronics would be guided toward the latter type of jobs. From my limited experience in my local area I believe that this type of jobs offers the best balance. Latter on you can move deeply into low-level development once you have beefed-up your resume if you so desire.
The trick here is to be identified by a competent manager that can match your education and experience to his needs. And you`ll be surprised how little managers need to know, technically speaking, to get a product going which means that you need to sell yourself well.
Senior positions depend heavily into the nature of the company. I find that importance given to hierarchy between employees is proportional to the amount of bureaucracy present. This translates into defined roles, which will need to be created (i.e., company is growing) or the position becomes available (i.e., because of retirement or lay-off). So for example, if you want to become an architect where you actually design a product or part of it; it will be harder in a big company. Smaller companies suffer less from hierarchy and give more chances to hold important and challenging positions but the pay usually lags far behind the bigger companies.
The best ways to get a job depend largely on the type of jobs available which are dictated by the companies implanted there. Where I am, there were several small companies that needed someone that could wear lots of hats. That`s how I started back 10 years ago. I had to manage a small network, program an 8-bit HC11 microcontroller, design the PCB, select the electronic components, do some soldering, do some mechanical design, and train the client and fix the bugs once the product was out the door. This path has led me to be a generalist rather than a specialist.
But I talked no long ago to the guy that was my boss back then, and he said that these days is cheaper to contract out the small jobs to specialized shops which wasn't the case back then or at least he wasn't aware of competent companies that offered a reasonable cost. So things do change. Be prepared to turn on a dime.
I think that you should not have too many problems going the embedded way. But you'll need to help the hiring person to match your skills to their needs. They may just assume that they need an EE just because they're working very close to the hardware. I think that you have a bright future. One of the most competent embedded programmers that I've met did a BSCS.
I hope I did not go ramble too much and loose you in the middle. Just one more thing. Remember that your worst enemy is yourself. Don't let self-doubt and/or laziness to get in the way. Control those and you'll have a good future.
[11] http://en.wikipedia.org/wiki/Field-programmable_gate_array