[Cavernio]

Aside: You know, just because something might actually have a yes/no answer, doesn't mean that it's something which we KNOW the answer to. Topics like that should totally be allowed in CT. (Like this topic.)

Anyways....
There's actually been a ton of research done in regards to handedness in psychology, (well, laterality specifically), so much that there's actually an entire journal devoted to it called Laterality. Left-handed or right-handedness doesn't just stop at your hands, and in fact the brain is actually wired differently depending on who you are. Interestingly, this phenomenon hasn't just been seen in humans; animals seem to possess it as well, as also in about the same ratio of 1/10 left-handed/right-handed as people do. It's even be found in fossil records where we see that trilobytes (ok, not sure if when I learned this in class if it was trilobytes or some other insects, but what the prof drew on the board looked liked a trilobyte) were found more often to have been biten or hurt on one side, indicating that they likely more often turned one direction over another.

As far as brain wiring goes, I forget/never knew specifics about what changes from where to where, but I know someone who's published a few papers which have involved differences in auditory processing and attention, where some people have ear advantages for processing language vs. pitch vs. meaning, which may be switched around depending on whether you're right or left handed.
I also, however, am pretty sure that if you purposely choose to change your dominant hand, your brain's not going to re-wire itself, (unless, perhaps, if you're really young, in which case 'choosing' may not be something you can even do), and that's it completely possible to change what hand you use if you really wanted to.

If you can't tell, I'm a strong proponent that showing a hand preference is something genetic. This by no means means that I think that people can't learn to use other hands to do things, or that people're strictly right-handed/left-handed. There's definitely a continuum of sorts for right/left handedness, and when studying anything involving laterality, handedness questionnaires involve a slew of questions beyond simply "what hand do you prefer to write with".

About being able to learn to write with your non-dominant hand as well as your dominant one later in life, I don't think it's quite as necessary as language to learn at a certain age. Writing is a fine motor skill, and we can learn new fine motor skills more than adequately at older ages. (take FFR for example.) However, there's still evidence that the people who perform at the top of any given fine motor ability start at pretty young ages; like, say, most professional musicians.
I'm assuming you're not planning on entering any hand-writing competitions, so I think you'll be able to learn to write with your other hand just fine, given enough patience and practice.