It does say at figure 7 that current limiting resistors are required but that is only for the 5082-7304 - it says so in bold above figure 7. In addition, all that part of the data sheet refers specifically to the 7304 - truth table, package dimensions etc all in one place.
The 5082-7304 is the '+','-','.','1' left hand "over range display" digit whereas the other devices are for standard numbers.
The 7300 has the decimal point at the right hand side, the 7302 a the left hand side and the 7340 doesn't have a decimal point - it has a blanking input instead - and does hexadecimal characters too. The 7300, 7302 and 7340 must be current limited internally.
Could it be the OP has used the 7304 in one (or more?) of the positions because when 'on' the two LEDs in series will have a voltage drop of 4v - and no current limit - and that's what's being reported.