I have some songs that are just about ready, now, to share with friends, but I am not at all thrilled with the super clean, harsh output of the sound. I find it not too pleasing to listen to my songs (at one sitting) more than just a couple of times, and it is because of that one issue. Might there be some software, or a plugin that I could get to, effectively, tame that?

Digital does not have to mean harsh. As you may know, I was a recording engineer in a previous life, and was working at this when 16 tracks went to 24 tracks, when vinyl was replaced by CDs, and along with many of my peers, we went through the rough transition... We all did, the first time you heard one of your projects burned to a CD versus pressed to vinyl, the cries of digital being harsh were raised.

I'll give you my take on it, and take it with a grain of salt, but it is based on first hand experience. I heard many engineers blame the technology, "...it must be the technology because I'm doing what I always do and it came out harsh."

I was still a young engineer at the time and took a different approach. I realized that all of the "tricks" I used to use no longer applied in the digital world. It was common to boost the high frequencies during record, even tweak the multi-track machines hotter at the high frequencies... The theory: in each transfer you lost a little high frequency. Working with high power, big $$ bands, who would spend months in the studio, the tape just simply lost that high frequency sparkle. So all these little tweaks to preserve the high end were in vogue. Even through to the mastering...

These "analog tricks" wound up making a digital project harsh. We used to "hit" tape hard, there was +24dB of headroom between 0VU and actual tape saturation. That's gone, the digital harshness you are experience probably comes from, "running through too many stop signs" along the way. If you are one who pushes the record levels to maximize it on the meters (instead of really listening to the sound) you might wind up with an overall harsh sound.

The use of Meters was different back then, in that there was an actual "noise floor" as a real thing. Here's what I mean, take a vinyl record, when you place the needle in the groove, you hear a pop, followed by the surface rumble (diamond on wax) that significant rumble is there all the time the record plays, most people can ignore it. But that the definition of noise floor. The given noise introduced by the media itself.

You tried to get your music signal significantly above this noise floor so that it became unimportant and could be ignored. Magnetic Tape, as soon as you placed that reel of tape on the machine and rolled it, there was a hissing sound, that's it "noise floor". The role of the engineers was to make sure of the signal-to-noise ratio (amount of signal stayed far above this noise level) without, at the other end, without distorting the upper end of the scale.

You left plenty of room so music had dynamics. I see people's mixes today and I can tell they are harsh by just looking at the silly ubiquitous waveform that seems to accompany everyone's song nowadays.... If it looks like a "stick of butter" you have someone who pushed the levels to maximize everything... In an attempt to make their mix LOUD, they boosted everything so it gets as close to zero as they can make it, then when they attempt to mix everything, it's harsh. (It's like a harshness build up).

What is everyone running from, it's not a "noise floor"; the noise floor induced by today's gear is microscopic compared to vinyl or magnetic tape... The level thing is OUT of CONTROL. If you see a clip indicator, most people don't even react. They just lower the overall output level and assume that if the overall signal doesn't clip it means there is no problems. But that ain't true. Clip the original recording and turning it down later does not fix the problem. You may not hear a horrendous crackle for a clip, it may only manifest itself as "harsh sound".

What metering are you using? How would you rate your metering habits?
If the signal is bouncing around between -24 and -12 do you panic because you think the level isn't hot enough? Be honest...
Transient peaks... Do you know what these are and how they impact your recording?
Do you record MIDI first, then render as audio? Are you aware this can introduce record level mistakes?
Do you realize that all frequencies do not record the same? Getting the same level on high and low frequencies might mean one is way louder than the other?
Ever wind up with the "too much cowbell" syndrome?
If your music uses drums and MIDI, what is the velocity of the 'average' snare drum hit?
Is your music compressed (not the leveling amplifier) but is everything packed to be near maximum... Is the first hit the loudest, does the mix have dynamics, does it look like a stick of butter when you create a wave file? If you can't see the dynamics, you probably will not hear them.

Harshness to me (my opinion) is most often a result of mis using the technology in an effort to be loud. Those small little peaks that clip (add up) they may not make a noticeable crackle but they add to the overall harshness of the mix.

Try mixing the project again but this time recognize maximizing level is not a race to make it loud, it started as a race to stay in front of the noise floor... The noise floor in today's digital recording is almost 'negligible' (that is a technical term for fuhgeddaboutit) no worries, on the noise end. The issue people have is at the opposite end where they compact everything up LOUD.

Harshness causes a kind of ear fatigue - it actually makes you tied listening to it. It is constant, without significant dynamics. If all your snare drum hits are 127, where and how do you create an accent? Answer: you don't. But if you establish 96-100 as your normal 2-and-4 snare bash, when you want to create a dynamic punch you have room to do so. If all your snare data is 127, where you gonna go? One way ticket to Harshness-ville.

Given just as my experience with digital and harshness, and how I learned to stop using analog tricks on a digital recording system... Hope it helps.
Remember, the VOLUME control is one of the few controls they give consumers. Making your mix loud - so it's as loud as everyone else's ( that's the excuse I hear most often) - if they like your song/music the consumer will turn it up. That they know how to do, make your mix LOUD they also know how to turn it down.