No - you cannot reasonably expect to do what you are trying to do.
You are attempting to draw more power from the transformer than it is designed for and MUCH more power from one winding than it is designed for.
I have 2 loads, first load 5VDC/2.5A put on EF and second load 12VDC/1A put on CF(DE shorted)
I'll take the VA rating as the same as DC Watts drawn - close enough in this case.
5V x 2.5A = 12.5 Watt
12V x 1 A = 12 Watt.
The 5V load is all on winding EF .
The 12V load is 50/50 on CD and EF.
So CD load = 12W/2 = 6W.
The EF load is 6W + 12.5W = 18.5W.
A transformer with two identical secondaries is usually designed to allow a maximum of about 50% of total power to each. You can unbalance them slightly but if you take all power from one winding you'll usually get extra losses and possibly transformer failure.
Here each winding is rate at about 13 VA/2 = 6.5 VA each.
You are trying to draw 6W from one winding (= OK) and 18.5W from the other (about 3 x overload).
Overall you are trying to draw 18.5 W from a 13 VA device ~= 40% overload.
You need a larger power rated transformer or smaller loads.