Wobble in port shaft

You would be amazed (horrors) how many issues come up from bad strap lifting. They try to splash real quick and correct when no body is watching.
So important to have pictures of the hull AND sling tabs marked.
I used to pick up boats at my fathers yard. I would crawl down into the hull and know where the shaft logs were. On a strange boat, with out any additional info, never a sling past that point.
I do not recall in the last several years anybody doing that.

Well, If you are not going to pull soon, You have to try.
A multi point will tell you quickly if it can be fixed or not.

If you do not have a bearing in the shaft log and only have one in the strut your alignment can be more complicated. If this is correct, your shaft is only supported at the two ends. So when you unbolt the coupling what’s going to hold it in place on that end for you to align your coupling flange faces to? The strut bearing is too far away and will not hold it in place. I had a 39’ Sea Ray sport fish that only had a strut bearing. Only way I could get it aligned was to haul the boat, unbolt the coupling, then make a plywood jig that held the coupling end in place while centering the shaft in the strut and the shaft log (seal slid forward) .Then I put the boat back in the water and allowed it acclimate for 24 hrs. Finally, we aligned the engine to the shaft with the jig holding it. I fought a vibration for a long time until we used this method. There are others on this forum way smarter than me so perhaps someone knows a better way to align it if you only have a strut bearing.

You have to figure the weight of the coupler and length of shaft from the cutlass less the prop and it's leverage (titter totter logic). It is a crude device but a large fish scale above the coupler holding that figured weight is fabricated.
I used to measure mean travel top to bottom of the shaft tube to verify this also (somewhere in the middle).
When the shaft was straight, I've done pretty good.
This is why you have to rotate the shaft coupler face and clutch coupler face in logical order, developing a truth table to determine what is in error. To further help hang calculations and where the shaft error really is.
Aligning a shaft coupler correctly is not an easy afternoon project and any shop that does two shafts within a short day cheated.
Ask for their truth tables and error findings. Dhu Du Du Dah,, what'sss that?? You just blew your money away..

I remember one job I was asked to fix. Previous tech did not realize a pilot bearing was in the log (or why the shaft seal was leaking).
You could not turn the shaft with a pipe wrench when he was done,,,,, in a couple of hours.
Island Packet 36 Motor express.
All day for me. Turned by hand when done.
Log stopped leaking.
I do have a clue...

Not directly related to your issue, but what type of shaft seal have you got, if I may ask?
Since you said you don't have any leakage, it must be a pretty good bit of kit.
Or an old school stuffing box, possibly?

Ref. engine alignment, while I still think you'd better pull the shaft first, if you don't have that possibility and you want to give alignment a try, I would firstly secure the shaft in its current position very firmly (with wooden chocks sustaining it, or whatever) before even start to unbolt the coupling.
This way, once you'll be done unbolting, you will have a first visual, albeit very rough, indication of if, in which direction, and by how much it is an alignment matter (or not).

Good luck, you'll need some.

Thanks. That confirms only that it ain't a stuffing box, though.
Most dripless seals are made as you describe, but among them there's the good, the bad and the ugly...
If by chance you've got a photo, it would be possible to check the exact model, also for your future reference.

You really need to try to figure out if you have bearings in the shaft logs or not. Without knowing this it will be difficult to know how to approach the alignment. I guess you could unbolt the coupling and see how much movement you get near the seal? Although if your cutlass bearing condition is unknown it might not be definitive. Also I’m guessing you do not know the I.D. of your shaft log as compared to shaft size? But even with a worn bearing I would think you would be able to tell. If you do not have a shaft log bearing you need to be careful that your alignment guy doesn’t align with the shaft sitting on or rubbing the bottom, inside of the shaft log. Trust me, this happens .
Maybe try a separate thread with a specific question on this to see if other owner’s might know if there are bearings in the shaft log?

Houston; there is a problem.
At idle, You have a bent shaft.

There are 2 similar threads on YF now.
Both are in denial they have problems.

That looks like more than .070 runout to me.

That would be wise.

You shouldn't even call that biting the bullet: a no brainer is more like it, based on your video.

BTW, I'm not 100% positive, but the shaft seal certainly looks like a PSS.
Which is pretty good stuff, as demonstrated also by the fact that you don't have any leakage in spite of the worst shaft wobbling I've ever seen.