Did you try rotating the gear 180° on the shaft to see if the holes would line up?  The holes are typically off center requiring the gear to be orientated a specific way on the shaft.

Lorena, Texas (South of Waco)

well, we installed the rebuilt distributor. It ran for about 15 seconds before the roll pin that connected the gear to the shaft broke in three pieces. When we tried to fix it, the holes didn't line up?? The roll pin pieces just fell out of the gear and shaft telling us because the holes didn't line up correctly, they installed a smaller roll pin which couldn't handle the torque. We sent it back and haven't heard anything from them in a week.  If it ain't one thing, it's two!!!

we're still here. Being lazy about finishing up the engine. It's 90+ in the shop so nothing going on until it cools.

4 two barrels. My partner had COVID so between crummy weather and him we will be on standby. He's working on a 54 Corvette at the moment. Anyone know where we can get an oil pan, let me know. The one in the car has extensive damage. June 20, got the oil pan repaired by a local pro. 60 degrees 2 days ago, 85 this coming Saturday.?????/

8 Webers or 4 two-barrel Webers?

That's one thing about being retired, you don't have to be in a hurry. We got the distributor back, is was their mistake. All fixed and installed. My partner and his wife picked up COVID somewhere and are hiding until his wife feels better. The shots have saved them from being really sick. We'll get the 368 running again soon. Dialing in 8 Webers is time consuming.

I hear you. I think the issue is when they modified the shaft to fit the Pertronix base unit the made the shaft too long. Because I had to send the unit back to get it to fit my block, the length of the housing might be a little off from the stock height which could have thrown other measurements off as well.  Eventually we will get this right. Their communication with me has been excellent. In no way are they ducking the issue and are standing behind their work. BTW, the hex shaft fits well. 

Sorry to hear about the additional problems.  Hard to see the internal 1/4" hex inside the shaft end but it is there isn't it?   From my limited dealings with Lincoln Y-block distributors, the dimensions are the same as that of the Ford distributor except for the gear.  Since it appears the gear is binding on the machined surface in the block, then it is not set to the spec of 4.991 to 4.996 shaft extended down as shown in the Ford manual. The thrust should be minimum .021" with .030" acceptable.  Also, make sure the internal 1/4" hex as mentioned above is indeed deep enough to allow the oil pimp drive shaft to penetrate as it should.  Is the camshaft installed in the engine?  If not, try installing a Ford Y distributor if you have one on hand to see how that fits in there.    As stated in one of my earlier posts:  I am at this time running a 1957 368 tach drive distributor in a 1964 Ford Y-block after changing only the gear. My point is the Lincoln Y and Ford Y distributors are dimensionally the same except for the gear. At least 1957 on. 

64F100 57FAIRLANE500

Well.... It was more than just the housing. I sent it back and got it back without the cap. Put it in the block and it didn't fit. The tip of the shaft where the oil pump shaft enters was wider than the hole in the block. I managed to take it out and set it up so I could use some 80 grit on the shaft while I spun it with my drill. Within 10 minutes I had the shaft sanded down where it fit nicely in to the block hole. Then the next problem..... When I bolted down the distributor to the block the shaft quit turning because the gear was pressing too hard on the block down below where the shaft tip connected to the oil pump shaft.  Back to FBO for modifications, again.  There are several lessons to be learned here, let me list them. Pic 1 is the hole in the block that was too small. Pic 2 is the sanded shaft to make it fit. Pic 3 is the bottom of the gear that bites down on the block when bolted down.
1-take plenty of pictures and measurements of your distributor
2-if you're the company that is making the changes, do the same before you tear it apart and modify anything.
3-never assume all the stories about distributors being swapped from one block to another are always true.
4-the 368 Lincoln/Mercury block is an ODD DUCK.
I don't blame FBO entirely for all the issues, my odd duck was something they haven't seen very much of. BUT, their lack of following #2 has resulted in some frustration that surely could have been avoided.
I'm also sure that they could do a wonderful job on a more common distributor.

Thank you. I sent the unit back on Saturday. He must have grabbed the wrong housing. My measurements on the one he sent me do not match yours, mine are larger. Thanks for the information.