i read a few replies to this thread so bare with my if my statement has been covered, but there are 2 different kinds of foam, the stuff youll find at walmart or a craft store, and also acoustic foam which is more dense.
acoustic foam will help you treat your room, but it is very expensive, if your handy at all, buy some rigid fiberboard (roxoul or oc703) and wrap it with a cotton cloth (just any one that you can get at your craft store and buy buy the foot.)
then fasten these to your walls, or make frames and baffles with them.
this will be cheaper than buying foam, and also this is what most recording studios use. well this and a combination of other things like diffusers, bass traps ect.
and im gonna say one more thing, your closet might sound boxy, but i have recorded in a 2.5 foot by 8 foot closet with sliding glass doors, and i had hideaway couch matress, foam (not acoustic foam) and heavy blankets/pillows covering almost everything in there. i found that my vocals were a bit boxy but not unusable, i just couldent put too much compression or the boxyness was very apparent.
when i slid open one side of my closet door, and hung a couple heavy blankets over that part my vocals got wayy better because they had some room to breath.
so yes, you can get good recordings out of a closet, but leave some room to breath, if your other room is big at all, this might be just what you want to do.
experiment with mic placement ect.
if you have a big enough closet, i would probably stand in the closet (as insulated as you can get it) and face the main room. i would then put a blanket a foot or 2 in front of the mic, that way you would be directing your vocals to the bigger room, and the blankets would help kill the flutter echo. this may be kind of hard to do, or require screwing some holes in your ceiling to hang the blankets though.
alot of people are gonna disagree with what im saying, but truth is, in most cases you can get better sounding recordings with blankets and matresses, than you can without treating your room at all. if you experiment around a bit with mic placment ect, i think you'll probably be able to do something similar.
the breathing room for your vocals is key though.