For salads I always use an olive oil based dressing. This will consist of ingredients such as lemon juice, garlic, Dijon mustard, balsamic vinegar, red or white wine vinegar, etc. depending upon the ingredients in the salad.
Sandwich fillings are a little more tricky. If it's the texture of the mayo you don't like, then adding to it to make an Aioli won't help much.
Once again, it does depend upon the ingredients used but even here it's possible to use a vinaigrette based on ingredients above. With this you can make a Muffuletto style sandwich.
Another possible option would be a Tapenade these can be made from a variety of ingredients to form a smooth paste, which can then be spread on a sandwich.