Most likely you just need to replace the two coils and your good to go. Just had some issues today with my Williams Jones Plugs as I had no power to flippers or pop bumpers . I already cleaned them but I pulled them all and re seated them again and all is well now. Just make sure next time you line them up right and take your time.

Before you order new coils, cut a wire of both coils, replace the fuses, and turn it on to make sure the coils are bad. It takes quite a bit to fry a coil in an EM. I believe the top fuse is 5v lights and the third down is 23v. Since the 5v circuit is also blowing, there is a short somewhere in the lighting circuit. I'd recheck your Jones plugs again just to be sure. Also, do a good visual check to make sure there are no stray wires shorting any of the wires on both male and female ends of the Jones plug.

Unlike a SS/DMD machine, you'll rarely if ever do any damage to an EM with a short burst of power like this. You might blow all of the bulbs, but the only coils I've ever fried are score reels and steppers when I left the machine on too long in that state (while troubleshooting).

Been there, done that. I once crossed the jones plugs on my Strato-Flite in the back box. Was rushing and not paying attention. Blew a but load of bulbs, a fuse and had a knocker firing off like a machine gun. Fixed everything back and after 3 years, I never saw any signs of damage.

You may not need to cut any leads on coils for testing, if you know how to do the alligator clip "battery test pull" method. Regardless you need to test contuinuity across the wiring on the machine up and down the chain everywhere you crossed circuits. Damage will be minimized, if you unplugged the game quickly.

I'll bite: what's the "battery test pull" method for testing coils?

Took me awhile, but I think I found it. At the very end, he takes the coil apart. Of course, I don't think I'm going to log a car battery down to my machines.

Check out this video on YouTube:

I can't see how this technique—as demonstrated in this video—would help obtain an accurate ohms measurement on any of the affected coils in @currieddog's Fun-Fest. He'll still need to de-solder one of the leads to determine if the coil's windings were damaged or not.

This is a "down and dirty" technician technique to determine initial game issues, not a definitive answer on coil, stepper, and relay winding damage.
It saves time when know which points require testing, and which do not when diagnosing an entire voltage schematic line when events happen such as this or due to short circuits. If an owner suspects damage to conductance and electrical resistance, further testing is still required. A car battery is not a recommended power source for testing pinball machines, as you do not need that much power. Cutting a lead and testing off of every coil on the line is a serious PITA, and generally unnecessary of you have a proper multimeter.

This tip saves time and effort while initially troubleshooting machines.

Bottom line:

If you are unsure about coil damage pull one wire off a lug on your coil and check with a Digital Multi Meter (DMM). I use one with auto ranging which actually lets me test a lot of coils without even pulling the lug wire. Most coils are bad if they are under 5 ohms however, there are a few exceptions to this rule with smaller coils, but this will pretty much cover bottom mech board coils as the smaller coils are generally located under the Playfield.

This battery thing makes no sense for a pinball. They are testing a "black box" in a car with the proper power source - a 12V DC battery. If you want to see if a coil is pulling in for a pinball, you could mimic this by simply hooking jumper wires from your transformer to your coil. You would probably only have to hook the hot line up, as you are probably safe in assuming your ground line (yes, I know, return line) is intact.

This sounded like a newbie that was about to order new coils without proper testing. I gave him a simple way to test.
As for testing techniques, yes there are other ways. In my opinion, this was the easiest to understand and do for someone inexperienced.

Check your wiring in the backbox in particular to make sure no wires are crossing anywhere. Could be something as simple as that. Also, did you have any real bad light sockets in your head before the incident? Could be a coincidence but a bad light socket that is shorted can blow a fuse with or without a light bulb in it. Maybe take out the Jones plug that goes to your backbox lights just to see if you still blow a fuse, if you don't, then your issue is definitely in your Backbox. Good luck in your quest, you are getting closer to solving the issue the more you look. Strong eyes and a bright light goes a long way in this business.

What was recently explained above has not been fully tested as recommended.
Please validate additional continuity for short circuits.
This means buzzing wiring that should or should not be connected, and identifying dead shorts.
Start simple, isolate, work forward.