Need Advice on Trans Problem

We put the boat in the water after winter storage. All was well except stbd gear oil temp was running a little high. Pulled the dipstick to find milky color in the oil. OK, so the cooler obviously developed a leak and we got seawater into the oil. Immediately drained the oil and refilled with new. Pulled the cooler and installed bypass hoses in place of cooler. Ran at dock at idle in gear until temp came up to about 140. Drained and re-filled again. No sign of color in the oil. Installed new cooler. Running the boat in idle at the dock and the gear is spinning in reverse. Not just a slow spin like normal but enough to move boat. Not like it is fully engaged but more than it should be. Ran in forward gear at idle and temp on stbd side running almost 30 degrees above port side. I'm guessing the added heat is from the forward clutch fighting the drag from reverse?
Any way to make adjustments to try and get that drag out of reverse? I now suspect the cooler was leaking last season and we may have sat all winter with contaminated oil. Could the clutches have absorbed moisture from the contaminated oil and swelled? Or are they metallic and may have got surface corrosion?
How big of a job is to rebuild these gears?
Donzi Z-54 with DD12v92TA's. Gears are Detriot Model 272A built by Twin Disc.
Appreciate any input. First trip scheduled for 9 days from today so hoping to get issue corrected in time for that trip.

Port was running about 108 degrees and Stbd was 142 when we shut down. Again this was at the dock, idling at 500 rpm in gear. Previously both gears ran about 140 at cruise and would get to about 150-155 if trolling for extended periods.

Water flow is good but it is getting warm. Confirmed temp difference with a temp gun to make sure it was not just a sender issue. Screen is being pulled as I type this to check for any obstructions.
We have been in contact with the Twin Disc dealer and a mechanic that knows these gears well. He says the clutch plates can absorb moisture and swell. Once we verify the condition of the screen we will go from there. Meanwhile dealer is checking on availability of parts but not sounding too promising for them being readily available.

Yes - straight 40w. I do not believe there is an internal filter besides just the screen.

I am fairly certain now that the cooler was leaking last season and we just didn't realize it. The trans did not give any indication of an issue until recent. But we had very few hours on the boat last season as I had to do a MOH of the Stbd engine in the middle of the season.

Had the boat taken out and ran for a good 2 hours this afternoon. Port temp was 152 and Stbd was steady at 180. Pressure good on both sides. Upon return to the dock the Stbd gear is no longer dragging in reverse.
We will check the screen again, change the oil again and do another test run on Monday. Technician says all numbers look good and to just keep an eye on it.
I still plan to rebuild this gear. Just hoping I can get thru most of our very short season and then do the rebuild.
I will update again after our test on Monday.

I plan to do a rebuild even if I am fortunate enough to get thru the season. There is clearly something gone wrong in the trans and at minimum I know sea water got in it. I will always wonder and worry about it if I left it alone. We'll see how our test goes on Monday and go from there.

Update: Changed the oil in the trans this morning. No sign of any contamination. At start up with trans cold there was a little bit of dragging in reverse but far less than previous. Took the boat and ran for 2 hours. Same result as previous. 150 degrees on Port and 180 on Stbd. Gear pressures within a few psi of each other. Upon return to dock no dragging in reverse.
So I will watch closely and see what happens as we use the boat this summer. Meanwhile we will get parts ordered and plan to do a rebuild at season end if the gear holds up that long.
In talking with the Twin Disc dealer we found out this is a Japanese built gear made by Nigato for Twin Disc. Supposedly a very good trans that are known to be very stout.

Thought I would follow-up on this. I used the boat for the rest of last season and had no problems with the Stbd trans. However, it continued to run about 20 degrees above the port side but still at the upper limit of acceptable.
Despite it working OK knowing that seawater had gotten in I knew it would drive me crazy. So I ordered new clutches and a seal and gasket kit. Parts are very hard to get for this thing so it took awhile but we got them. We pulled all the gingerbread out of the way and separated the trans from the engine. The housing is too large to fit thru the engine room access stairs so the work is all being done in the engine room. Although no major corrosion was found you can definitely see signs of the saltwater. We tore everything down and took most parts back to the shop for cleaning and inspection. Fortunately all seems OK with the exception of one bearing on the input shaft. All the clutches have been replaced and we are reassembled as far as we can go. We located a replacement bearing and it is en route. Once we get it all back together and tested I'll post a final on results.

The Twin Disc Rep that we used said these gears are as bullet proof as they come. He said that is one reason it is hard to get parts. The demand is so low they quit stocking them. If not for the leak in the oil cooler I doubt I would have any issues.

On mine, the oil cooler is last in line before the water discharges the boat. We found coolers that are nearly identical. Same diameter and length, same inlet/outlet sizes and very similar tube bundle. Only difference is slight change in taper in/out from main hose connections. We replaced the coolers on both Port and Stbd.

If memory serves yes. Although now that I think about it I believe there is a small H/E for the power steering system after the gear cooler. But I can't recall which one is in line first.