Some of the shallots, red onions, yellow onions at this time of year are a bit strong. They've been in storage for months.
Overpowering. When the new crop comes in from the Southern Hemisphere you should have better taste--which is it now...along with texas sweet onions..and vidalias.

In the mean time.The large "spring onions" are in season (the ones with the big bulbs on the end)..they are sweet and mild.

If you let the chopped shallot soak in a little vinegar for 15-30 minutes, it helps soften the sharpness of the flavor.

Agree with fiiveandspice. I try to make the vinaigrette an hour or so before eating and the shallot mellows and changes character.

Pickle in a little lime juice, kosher salt and sugar and leave to 'macerate', rest for 10 - 30 minutes

CI recently tested how to make raw onions have less bite. They found the best method was a 15 minute soak in a baking soda and water solution (1 T b.s. per cup of water.) The onions need to be well rinsed before using them. It works for onions because the alkaline b.s. neutralizes the sulfenic acid found in the onion. Maybe the same would work for shallots.

I'm not a big fan of raw shallot or onion either. I usually substitute scallion or chives.

I think cutting back on the amount of shallot you've been using and letting it mellow in the vinaigrette might help too.

But, I also only use shallot for salads that it enhances. For instance, an arugula salad, or the roast potato salad I made last weekend, can take a good wallop of shallot and Dijon in the vinaigrette. If it's a delicate salad, like soft Bibb lettuce, I just do a very simple one, and leave the shallot out (just evoo, vinegar -sherry or wine - and a touch of Dijon and S&P) so it doesn't overwhelm the greens.

(And sshhh, don't tell the food police...but I often also add the teeniest pinch of sugar - not enough to create any sweetness at all, just a smidge to round out the flavor a little. Learned this years ago from my old neighbor - an Egyptian who grew up in Paris and was to this day the most talented cook I've ever known. Miss her.)