I'd be afraid to cleat the sheet. I'd rather get tired fingers than capsize, but I share your suffering.

In light to moderate breezes I like to use a "wrap cleat" as shown in this brief video:

https://www.youtube.com/watch?v=n9Ps9TlSnwM

A single wrap around the tiller extension takes away nearly all of the work of holding the sheet and allows one to have a hand free. In this case you hold both the sheet and the tiller with one hand and can quickly dump the sail by sliding the wrap off the end of the tiller extension.   

   Possible solutions:

Add more mechanical advantage by adding another "up-down" between the boat block(s) and boom block(s), many ways to rig this.  It might require more or additional blocks, or a "fiddle block."  Remember that each additional turn of line means that you have to move more length of line for any given amount of boom movement.  And rigging it from the center of the boom to the centerboard trunk is an option that requires less total line length than rigging from the end of the boom.

Subsitute a ratchet block for the mid-boom block.  Ratchet blocks only freely rotate in one direction (trimming in on the sail).  When easing the sail the block doesn't rotate, the line must run out against friction.  Most ratchet blocks are selectable - meaning you can disengage the ratchet so that the sail will ease out easily in light wind.

Put in a cleat or cleats for the mainsheet.  Remember that a cleat on the mainsheet in a dinghy can lead to capsizes in gusty conditions, but skill and care can (almost) always obviate this hazard.  I recommend a "jam cleat" - one horn of the cleat is standard, one is a wedge that holds the line when wedged under it.  Very much easier to undo quickly than if you'd actually finished a securing turn on a double-horn standard cleat.  And with either a standard cleat or a jam cleat, you rarely need to finish the cleat turns at all, just the friction of running under on horn is usually enough to serve the purpose.  With the layout of the NE dory, I recommend a cleat on each side of the center seat.  This keeps the cleat from interfering with your butt while rowing.  (The back of the centerboard truch isn't an option for a single cleat for the same reason - a cleat would kill your back.) So each side of the trunk is an option, but that means you have to reach in towards the center of the boat while hiking out, and because you're then doing two cleats anyways, why not go out to each side of the seat, where the cleat is more handy on each tack?

So, I first recommend the ratchet block, you will probably find that solves all problems with nothing else needed, and is achieved by simply buying one piece of hardware and no new screw holes., and no change in mainsheet lenght.   I have a ratched block on my boat.  But a ratchet block alone does always require one hand to hold the mainsheet no matter what, and it is convenient to sometimes not have to do that - when needing a drink, putting on the chapstick, whatever...  So I also have a jam cleat on each side of the center seat.

Finally, if you can, either look online ( all boat "class" websites usually include a rigging description page) or wherever small boats are sailed in your area take the time to visit with a dinghy sailor (racer preferred) for suggestions.  They've solved the mainsheet problem a dozen different ways on a dozen different types of small boats.

Thanks both of you, these are great ideas! I like your way of thinking. Don't need a cleat if we get rid of the underlying problem which is the stress on the sheet.

Not sure wrapping the sheet on the tiller would work though since the person controlling both of those (usually me) is on the stern thwart with the crew on the front thwart. We've tried switching it up, but I prefer to face forwards.

Cleats, if any, could be aft, but Plan A is to get a ratchet block.

Anyone reading this who, like me, didn't know what a ratch block was, here's an excellent video on YouTube The Ins and Outs of Ratchet Blocks by Harken.

My schooner had 2 ratchet blocks, one for each sheet, while my Faering Cruiser has a 4:1 block and tackle on the sheet. Both systems solved the problem and I'm not sure which I prefer.

The ratchet blocks were a bit pricey (Harkens) but excellent quality and natural to use. I could control 2 sheets with one hand without feeling as if my arm and hand were overworked. Since the NE Dory only has one sheet the price is halved, not to mention CLC having a more economical alternative to Harken now, so that issue goes away. The sheets could also be led to camcleats, if desired. Those allowed a secure line when needed with instant release.

The 4:1 tackle is also very easy on the arm, but all that extra line has to go somewhere. There's room in the Faering Cruiser, but in a smaller boat it may get to be a snakepit, to borrow Bolger's perfect description.

Laszlo

Pinch the sheet under a toe against something.  Less likely to lose your dentures.

.....Michael

   I have had clam cleats on each side of center thwart for quite a while now. Under light conditions it is great as I can use both hands for other needs. When the wind is variable it works to keep the strain off while I hold the line ready to lift to release. In higher winds I have used it briefly, ready to let it loose easily, to get a drink or zip a jacket...