I'd suggest taking the cabinet apart - you will get much better results in the end by doing that. Otherwise you'll end up with bits of old finish peaking through the cracks. This does not happen if you have a cabinet that was origiinally finished assembled - but that is not the case with this one.
BTW - you have a lot of cleanup to do before you even consider sanding - otherwise you'll end up with a very uneven surface and possibly sand right through the veneer.
This is the older style of your cabinet (this one is from early 1920's) - yours will come apart about the same.
Here is a neat way of dealing with having a lot of pieces - get some different height disposable cups, and you can layer your pieces for drying. This is my quilt frame.
Canning jar lids make excellent woodworking "cookies" That is what I use for the bottom layer supports.