It looks like the real complication is that you have 17x4 (68) BCD input signals. If you drive the displays 1 digit at a time that will require 17 digit select signals + 7 segment signals. So it sounds like you need a device with about 100 pins.
The only devices that I can think of off the top of my head that can nearly manage the task are CPLDs and FPGAs. (They won't be able to handle the current coming through the digit's common cathode.)
The (now obsolete) Xilinx XC9500-series CPLDs and some transistor array ICs (like the already-mentioned ULN2003) could do the job. The XC9500s are 5V parts that come in fairly high pin count packages and support high current (24 mA) outputs.
You might have to resort to getting the parts from ebay or similar sources if you go that route, however, since they have been discontinued. Then, of course, there is the issue of programming them. Which is to say that you'll need a programmer (about $50.) So you might keep looking for other options and use a CPLD as a last resort.
If it were me I'd probable split the task up between 4 XC9536 parts in 44-pin PLCC packages or 2 XC9572 parts in 84-pin PLCC packages. These are surface mount parts but are very easy to solder. So far as driving each of the common cathodes goes, the ULN2003 would work. You'd need three of them in this case.