Hm.... I suggest you to use of AA batteries first if it works fine without problem, then go for bridge before the 5v regulator replacing the AA batteries. First eliminate the doubt on MCU.
Follow these steps for Adapter power supply.
1. Use a bridge like W04 or 4 x IN4007
2. Give the output of bridge to 7805 couple with capacitors before and after regulator.
3. Measure the voltage output of 7805 without MCU connected.
4. Just connect MCU no other components including the Crystal and associated caps.( Here I think it should not heat because there is no load).
5. After observation of heat then connect the crystal and caps.
6. Finally connect your ULN2003 ( I suspect this one now, "a silent killer". I remember my washing machine stopped because of faulty ULN2003 it also damaged a CAP after 7805).
I think one of the above should solve your problems and should function normally once you find the faulty components.
Thankyou
---------- Post added at 07:44 ---------- Previous post was at 07:43 ----------
if you find heating problems even with AA batteries replace ULN2003.