PaulRB:
Yes. It is a 12V strip, the strip can be cut after every 3rd led because trios of leds are wired in series, with a resistor to limit the current to 20mA. So each segment of 6 leds should draw 40mA, and a digit should draw 280mA when displaying "8".
How did you interpret the OP's description of the circuit, Dave?
based on the sketch the OP posted,
each LED uses 20mA, x 6 = 120ma per segment.
a 7 character digit would use 7 x 120 = 840 ma
but before I start offering that a typical ULN2803 package [ channel ] can handle about 500mA, I would want the OP to post exactly what strip he is using.
superbrightled rates a 5050 led strip with 60 LEDs for 808mA for 60, or 80.8mA per group of 6
80.8 x 7 is 554 ma and [ not close to the threshold of overheating the 2803 package if they are all on at the same time.
assuming a scoreboard showing all 0's is only 6 segments, x 80.8 = 498mA,
both cases would lead to failure over time, exactly as the OP has listed. [ only valid if this is a package not channel ]
if the strip properly attached to a heat sink ? what happens to that exact LED when it over heats ?
generically, just using larger drivers can fix some problems, however, I like to do the math using the data sheets.
[ did we confirm the ground between the 12v and the Darlington ? ]
[ max out of the 74HC595 is 70mA, ]