You're probably not harming the 344 by letting the needle mechanism work while you wind bobbins, but you're not doing it any favors, either. Essentially you're wearing out the main sewing parts twice as fast by having them work while you wind as well as while you sew.
I think you should get out the new machine. I bet you can just plug it in and sew with it - take it out of the box and see! It probably also uses the same bobbin as your current machine, so if you can figure out it's bobbin winder, you could use it to wind and keep using your 344 to sew.