The problem you describe occurs on back to back wins on cards 1&2, 3&4 or 5&6 or if as you say wins on all six cards. Look at switch E on the Replay Cams Index coil. The only time Switch E should be closed is when the coil is fully energized. When the cams are in the roll-over position (which is where they are shown right now with the power off) Switch E needs to be open. I've attached a picture of what this should look like. Notice Switch E is open in the photo. The cams have been stopped right after scoring 4 replays by shutting the game off immediately after it had scored 4 replays. This is the ideal position to adjust the switch. The blade that needs to be adjusted looks will likely be the one with the blue w/yellow tracer wire. This is important because when the cam returns home switch E will be open and appear normal. So, by stopping the cam by shutting off the game after 4 credits are registered, the Replay Cams are neither Home nor Energized (They are in the roll-over position). If this switch is closed while the cam is in this state then this is the problem.

Switch E Search Index Coil (resized).jpg

The problem you describe occurs because the machine cannot score back to back wins on cards 1-2 or 3-4 or 5-6 and that’s exactly what your Dixie is trying to do. So lets say you have wins on cards 1&2. It will score properly on card 1 and then step up the Replay counter on card 2 without adding any credits to the score meter. Look at switch E on the Replay Cams Index unit. The only time Switch E should be closed is when the coil is fully energized and should be open not only in the Home position but ALSO in the Roll-Over position. I've attached a photo of what this should look like. Notice Switch E is open. When the cams are in the roll-over position (which is where they are shown in the photo) switch E must to be open. To take the picture, the cams have been stopped right after scoring 4 replays by shutting the game off immediately after it had scored the 4th replay from a three-in-a-row win. This is the ideal position and way to adjust the switch. The blade that needs to be adjusted will likely be the one with the blue w/yellow tracer wire going to it. This is important because when the cam returns home switch E will be open and appear normal. So, by stopping the cam by shutting the game off after 4 credits are registered, the Reply Cams will be in the roll-over position. If switch E is closed in the Roll-Over position then this is the problem.

If you were to place the balls in 14, 15, 20, 21, & 22 you will register a 3-in-a-row win on all six cards. When you start the Search Cycle, a properly working game will score on cards 1,3 & 5 and then stop. You will have to start the search cycle again to collect on cards 2,4,& 6. The game cannot score back to back wins on the same search unit (1&2, 3&4 or 5&6) during the same Search Cycle. Switch E on the Replay Cams Index is responsible for this. If the switch is improperly set (Closed during the roll-over portion of the cycle after scoring a win) then the game will attempt to score the next win as it normally should but it will have "Run out of room" due to mechanical limitations. The result is that it will step up the next card unit but the player will be "cheated" out of the replays as the machine cannot register them at that point. Switch E should only be closed when the coil is energized and then the game will register back to back wins properly, albeit on seperate search cycles. If not, the game will act as I described.

If there is no Alternator Unit it would have had to have been removed somewhere along the way. You would think that some cut wiring would be evident. You'll need it to properly score the win when successfully going for Double or Nothing while on a Double Double hit. It's hard to say if the unit was just taken for use in another game or some modification has been made. So what if some mod has been made....
I'm curious, when you play all eleven coins and drop a ball in the magic number do you get the Double lit on four cards and Double Double lit on two? Then if you get a win on a card with Double Double lit then does the machine only register Double instead of Double Double or register improperly? If yes:
1. Start the game over and set it up as above lighting the Double Double feature and before collecting the win check that the Eleventh Coin Relay is energized.
2. Collect the win.
3. If game pays only double or otherwise incorrectly, check to see if the Eleventh Coin Relay has disengaged.
If you've gotten this far then you likely have a shorted diode. Check the diode on the Double No. 1 Relay (page 36 in the manual).

It is located at the far upper right corner looking into the back of the machine on the light board right behind the 1st card light. Baldtwit's explanation is right on. PM me and I will send you one if you haven't already found one. Either the whole unit or just the cam, whatever you need.