You may have to carefully hacksaw the old drain fitting out.  Most likely it can be replaced with parts from a plumbing supply store or a big-box home centre; with maybe some creative adoption pieces to get it eventually to a drain hose.

One useful way to get more access to the undersink area is to remove the 3-drawer unit that sits to the starboard side of the cabinet (facing the nav table).  There are maybe 8 screws that hold the entire assembly in, covered with teak bungs that can be carefully drilled out.

Richard, I recently found my sink drain leaking due to broken plastic nut. You can get the original part for about $25.00. It is manufactured by Forespar Marelon. I will place more info with photo by Monday. There is hope.
Stay tuned
Haro.
More info added:

They are Forespar Marelon Forespar 907017 90° 1"I.D. Sink Waste Drain,  
Forespar 907017 1 Id Sink Waste 90 Deg Sw-4
Features: 90° configuration 1" I.D. tubing 2-7/8" flange 2" cut-out
Forespar SW-4 Sink Waste / Drain

Available from
   https://www.boatersland.com   866-828-1033    $ 24.99 + Freight   
   https://boatplicity.com              888-854-6152     $ 23.86 
   https://www.defender.com       800-628-8225   Item # : 501267    800-628-8225     $ 28.99 + Freight

We had a similar problem 4 years ago when the boat was new to us.  The original chrome plating on the plastic drain fittings had corroded through and was causing a leak. (See port side drain in attached photo.)

Everything came away fine from the stainless steel sink cleanly, but as I was in a hurry instead of ordering the proper marine fittlings online, I just wen to Lowe's/Home Depot and bought the fittings you'll see in the photos in the link below.  Cheap, easy and works well.  As these are drains there is no issue with pressure in the lines.  An anti-siphon might be a good idea, but none was originally installed and I believe this was the reason for the original very deep sinks?  Extreme heel to port might bring some water into the port side sink, but the sinks are fairly close to the centerline of the vessel. 

I did remove the screw bungs and took out the drawer assembly.  To ensure a seal without the proper sized washers I used Auto Goop around the sink on top, stick-to-itself plumber's tape on the threaded shaft of the new sink fitting and cut off the plastic disk to make a collar that would press up against the underside of the sink, then reused the plastic elbows and reassembled.  Remember that the down-pipe below is below the waterlilne!

Link to pics:  https://www.flickr.com/gp/47932873@N00/06u5z9

I did look into getting a new single shallow sink, but the only version I could find online (see old thread re: "new galley sink" in the forums) was too expensive at $350 or something like that.  The job looked a bit daunting to effect the swap and the Admiral said she was fine with the original twin bucket sinks.  I would likel to eventually change out the faucet for a one-handed version.
 

I too want to swap out the dual knob faucet in my galley sink for a single handle with a sprayer head. A couple questions arise:

1. Is there something 'off the shelf' at Lowes or Home Cheapo that would be durable enough for the marine environment and withstand the freezing temps of layup in New England? Or better to spend the $ on the kit from Catalina or CD?

2. I assume removal of the stbd facing drawer assembly would also be key to accessing the base of the faucet? Any other tricks?

Thanks again

Mike

Mike, as most faucets are made to withstand exposure to a certain amount of moisture and are all made of chrome-plated brass and plastic, I'm not sure that most marine faucets are much better.  Size might be an issue, but the big box hardward stores have a vast range of different sizes. 

Before removing bungs and unscrewing much I'd suggest trying to use a flashlight and mirror through the big drawer opening under the sink to see what might be in the way.  I recall there is a plywood partition running down from the countertop behind the starboard top drawer that might be in the way of accessing the back of the sink on the forward side of the galley.

I realize this thread is a few years old.  Thanks for the pics Rich.  I have a 1992, MK 1.5 where the sink drain is approx 1/8" higher than the bottom of the sink so there is always standing water in both sinks.  If the sink wasn't sandwiched under the countertop I'd replace it because the sink is now pitted from the PO allowing water to sit for days/weeks.  If I do the drain replacement shown here will it lower the drain enough so water leaves the sink fully?  

Randy, you can buy one Lowes version and if it did not fit, you can return it. I suspect someone installed your current drains incorrectly. I am pretty sure if you remove your drain you may be able to reinstall them properly so they are flush.
Haro