700mA is definitely a problem. The entire circuit including the WiFi module should draw no more than about 300mA.
That discolored solder is on the 3.3v regulator that powers the ESP. Is the regulator the thing that's getting hot on that board or is that discoloration possibly just flux that didn't get cleaned off?
What happens if you plug the interface board (the one in your picture) into the clock but don't plug the ESP into it? Does the current stay around 260mA or does it jump up?
Thinking back a long time, I recall I have had 2 ESP-01 modules go bad in a similar way: extremely hot and would not work at all. Both happened on the same day. I was working with them on a breadboard with 5v and 3.3v rails. At the time I just thought that I'd hit them with the wrong voltage on their Vcc pins, replaced them and forgot about it. You might want to measure voltage between Pin 8 and Pin 1 on the 8-pin connector with ESP unplugged from it. If it is over 3.4 volts then the regulator on the interface board is bad.
In case you need it here's a ESP-01 pinout I recently posted for someone.
www.tubeclockdb.com/component/kunena/12-...m.html?start=10#9157
Ignore the photo, scroll down a bit to the diagram.