This is a tall order, but not impossible
Our reference is this description of Titan's atmosphere:
Like Earth's atmosphere, Titan's atmosphere is largely composed of molecular nitrogen, but it has only small amounts of oxygen and water. Instead, the moon's clouds, lakes and rain are made up of hydrocarbons — molecules composed of hydrogen and carbon, such as methane and ethane. [Latest Saturn Photos From NASA's Cassini Orbiter]
The huge amount of methane makes the haze on Titan "like L.A. smog on steroids," explained Scott Edgington, Cassini deputy project scientist based at NASA's Jet Propulsion Laboratory in Pasadena, Calif.
Titan's nitrogen and methane molecules are broken up in the moon's upper atmosphere, about 600 miles (1,000 kilometers) above the surface, by sunlight and particles from Saturn's magnetic bubble. The positive ions and electrons that are left over trigger a chain of chemical reactions, resulting in hydrocarbons — including PAHs, which are large carbon-based molecules that form from the accumulation of smaller hydrocarbons, researchers say.
PAHs can keep building up, aggregating to form bigger, heavier molecules that sink lower into Titan's atmosphere, eventually leading to the aerosols embedded in the dirty haze that cloaks Titan, closer to its surface, below 300 miles (500 kilometers) altitude. (Source)
It's important that you understand that third paragraph. It isn't just the methane that's the issue, it's the sunlight that's the issue. We can simplify this with an analogy. The methane is "gasoline" and the sunlight is the "automobile." Without the sunlight, you have a foul-smelling atmosphere, but if we ignore just how small the oxygen component is (and this simplification is so outrageous it's making angels weep!), its presence isn't intrinsically lethal (you should be thinking, "yeah, like sucking helium from a balloon, you don't just die from it..."). Hopefully you get my point.
Anyway. You're basically asking for a way to have the aesthetic of Titan's atmosphere without having, well, a trillion humans putting around in mobile combustion engines.
You're problem is that the methane-sunlight reaction (aka, the "automobiles") happens at the top of the atmosphere, where here on Earth the automobile reaction happens on the surface. All the particles are almost-settled to begin with rather than being created like a permanent acid rain. But that doesn't make a suspension-of-disbelief opportunity impossible.
One more thing, I'm assuming that you don't care about a haze through the entire depth of atmosphere, but really only in the first 1-3 miles. If you want the whole atmosphere, you need a top-down solution like on Titan. I'm having trouble coming up with a solution like that without the toxicity.
One method is to have a very warm water planet. Such a planet, combined with the nasty winds that would occur thanks to the sun-side-dark-side cooling cycle, could believably have a permanent fog. You could also have a cold water planet with a lot of underwater thermal venting, so that the water is warm and the atmosphere is cold. Even on a still day, you have fog. If you think about it, our planet has a LOT of days that would be exactly what you wanted if the clouds where near the ground rather than 0.1-5 miles above ground. You just need them to not float so much... which might be one of the reasons why 0.14G Titan might have an advantage over 1.0G Earth.
Another method would be to have the opposite, a nearly dry planet with almost no precipitation. And make it a bit warm. Thus, winds... thus, dust. But those winds would need to be constant. Dust is pretty heavy. No wind, no "haze."
A third solution would be something like a jungle planet with ubiquitous high ground water but almost no standing water — and some serious pollen-creating plants. Habitable, if you have enough antihistamines.
A solution that doesn't easily fit your criteria would be a highly volcanic world. But probably too toxic.
Or... you could have a solar system where a massive planet is being torn apart by the sun, resulting in a trail of dust/water that falls onto your planet's atmosphere. Technically, that's a non-toxic top-down solution — but keeping the side of the planet opposite the trail of incoming dust with a permanent haze might be tricky. Allowing the giant planet to disintegrate for enough time, and yet fast enough, that it doesn't matter what the orbital orientation to your habitable planet is might also be a bit tricky. Still, suspension-of-disbelief.
Perhaps your biggest problem is that you must have that nitrogen/oxygen mix to be habitable. You can add a lot of water to that mix and still be habitable, but I suspect almost any other element or molecules would be in some quantity, toxic.
Hopefully this answer will jog the imagination of some others!
