There is no "best" time to mulch nor is there a wrong time to mulch, you do that when you have the time and material.
Fresh grass clippings can produce foul odors and that is valuable nutrients escaping to the atmosphere. What might you have that could be added to the clippings that would help prevent the odors?

Oh man, I've been frowned at by neighbors many-a time too. The trick is to keep at it, make it work, so they eventually see you're not crazy, it looks pretty good, and you did have a plan and reason for everything you've done. I don't till either, and have been not doing so for decades, with great results. Whenever you disturb the soil, you wreck the natural layers (that may not be visible to your eye) and drainage, as well as possibly killing-via-disturbance-and-displcement many of the beneficial microscopic critters that live there, doing their best to constantly fix the soil. Eventually, the ground will be come spongy, springy, dark, moist yet never muddy, fertile.

"I mulch by laying newspapers on the beds and then cover them with good grass clippings every April and then plant in May."
Smothering with paper or cardboard is usually a 1-time thing. Once you get a bed going, then add mulch, leaves, grass clippings, small yard trimmings, finished compost, a weed that you pulled and left baking upside down, discrete kitchen scraps (scrape back the surface, dig a little hole, bury & recover the top layer, if you don't have a compost pile or bin.) Smaller amounts of varied items more often is great, and offers a wider range of material for the tilth and fertility of your soil.

This keeps the activity going, which is necessary to have a 'live,' healthy soil. A short video lecture that should help a lot.

"The grass can smell and those who live next to me give me ugly looks at times. Am I doing this wrong"
The odor of anaerobic (without oxygen) decomposition is not necessary, and a sign of an imbalance of a predominance of greens, with no browns to absorb the moisture. Are the grass clippings sitting in a bag or other enclosure for a while before being spread? That would create a lot of odor fairly quickly. Putting it around the beds right away would prevent that, by allowing oxygen to reach the material. Mixing it with dry leaves, shredded paper (neighbors frown again,) or making more of a lasagna with it and the layers of paper, would all help you get more bang from your material, without odor. Lay a little grass, a layer of paper, a little more grass, more paper, so they can react with each other and the air to decompose aerobically.

" or should I mulch deeper in the fall and let it set over the winter?" Absolutely. Fallen leaves are a no-brainer, mixed with the last of the grass from the mowing them up, they will disappear especially quickly, no risk of them matting. THOUGH, matted leaves suppress weeds well. Some rake them onto beds whole to do this very thing around shrubs that don't need to poke through the surface to get going in the spring, they're already above all-that. With many kinds of leaves, it's nearly impossible to put enough without it looking like a compost pile, that they won't disappear completely by spring. In OH, 18" of unshredded ash, maple, cottonwood leaves would be completely...

While some do believe that newspapers are a 1 time thing they can be used often and will aid in extending the amount of other mulch material you have, allowing the gardener to put down a 2 inch mulch layer instead of a 4 inch layer.
Because of the C:N ratio and moisture grass clippings have piling a bunch of them up will allow them to be digested rapidly but much of that Nitrogen will escape to the atmosphere and that is that strong, and disagreeable, ammonia like odor given off a pile of grass clippings. Mixing something with a high Carbon content, shredded leaves for example, will help contain that N and keep the odor down. In my experience grass clippings only very rarely go into anaerobic (in the absence of air) digestion and then only if piled quite high. But a layer of 2 inches could produce that strong ammonia odor of N escaping to the atmosphere.