Mark,
Let's see what I can find.
Transmission
Most '63s used a Warner Gear 4-speed. '63 used an aluminum case. On the right side of the case is a casting # of T10D1. Next line is "W.G.DIV" Casting date is in the upper center of the rt side of the case. Date is made up of month by letter. "A"= January and so on. Then the day of month and then the last 2 digits of the year.
The year may be omitted on some.
Late in '63 the Muncie transmission started showing up. This '63 trans is unique and differs in many ways from the '64 and newer. There is an exposed area of the machined surface at the lower rear where the side cover attaaches. The assembly code is located there. It is a stamping made up of the letter "P" followed by a date code for month and day.
On the right side will be a casting # of 3831704, the words "PAT. PEND." and General Motors Corp. In '64 the words "U.S. Patent No. 3088336" were added and they will be on all '64 and newer Muncie cases no matter what the casting #.
3885010 may be a newer Muncie if it has the Pat. No. It could be a Super T-10 also. I confined my comments to identifying the '63 transmission main cases as we are only concerned as to if yours is original to your car.
Differential
The rear carrier was used over other years so the best way to ID it is to check the stamped code on the bottom rear of the carrier. There is a 2 letter ratio code followed by the date code for month, day, year. It may take some scraping and brushing to find this.
There are 9 different codes based on ratio, posi or not, and trans. An example of what you are looking for would be: CB 11 29 62. This is a 1963 3.36 positraction manufactured on Thursday, November 29, 1962. Post your code or PM me when you find it and I'll look it up for you.
If I can heip further let me know.
Tom