Visualizing harmonies takes place in what I and others call the aural image; basically, being able to hear music internally with little or no outside stimulus. When instrumentalists practice this, they usually do it by singing, since the voice is that much more closely related to the brain than any given instrument. For all of the above, though, it can be done internally as well.
A lot of what will make you better at this falls within the realm of ear training. I would start by aurally visualizing songs that you know, without singing at all, and see if you can get BOTH the melody and the bassline/chords going in your head at once. (i.e. "Sing it in your head." NO CHEATING by whistling, humming, or whatever. Yeah, it might help at first, but it'll be a crutch you can't rely on in other situations.) In many cases, harmonies are the result of these two factors: the bassline (and chord) tells your ear which notes sound "good," and the melody gives you a contour to sing next to.
Think of the melody as a guide, for which there are invisible harmony parts both above and below that form a chord defined by the bassline. The way I see it, if you have internalized both the melody and the chord structure, you have enough information to sing a harmony part. You can practice this by starting out on the melody line, and then try jumping up or down to one of the "invisible" harmony parts. "Look before you leap," as in, try to have an aural concept of where you're going before you make the jump.
I have never covered one ear while singing harmony. As far as I'm concerned, that's a fad invented by record companies to make oversouling starlets look like they know what they're doing (i think they just look silly). Music is a hearing art, and if you are covering one ear you're only making it harder for yourself to tune and hear the chords that are happening in the music. If that's a sensory overload that's too much for your ear to process, you should go back to working on internalizing the melody and harmony: they should sync up, not conflict with one another. If you don't know the melody and harmony already, the music happening in the moment is your only reference (outside of typical chord progressions and melodic contours defined by our culture). Master improvisers live in this space. Keep that in mind: really, what you're doing is improvising a harmony part. Any practice you do improvising in general is only going to help this skill.