Are you absolutely sure about the year? Because it sounds a lot like “Everybody Hurts” by R.E.M. but that was from the early 90’s.
If it’s not Take Me Home, then the only other thing I can think of is the entire Alice Cooper album “From the Inside”, which is, literally, about his experiences in a mental institution. I realize that the album was released at least a decade too early for the OP, but I still here some of it getting radio play. I came to appreciate Alice only a few years ago. I think he’s WAY underrated.
***Solsbury Hill ** * by Peter Gabriel?
This is the song that sprung to my mind. Here is a link to a clip of the song (WMP). I remember it being played often back then, on stations in Saint John, NB, and one in Calais, Maine. (K100, C98, and WQDY). Which is pretty close to being in New England. 
Wow! Some good suggestions.
It’s definitely NOT Duran Duran’s “Rio” (please… :dubious: ), Phil Collins’s “Take Me Home,” Peter Gabriel’s “Solsbury Hill,” David Bowie’s “Ashes to Ashes,” or anything from U2’s “The Joshua Tree” album. Having just listened to them, it’s also not Yaz’s “In My Room,” Suicidal Tendencies’ “Institutionalized,” or anything from George Harrison’s “Cloud Nine” album.
I haven’t been able to find an easily-accessible online source (iTunes, Amazon.com or Barnesandnoble.com) for The Smiths’ “Last Night I Dreamt…,” but will try to do so. The lyrics to Bourgeois Tagg’s “I Don’t Mind at All” look familiar (esp. the last paragraph); it could very well be that one. I’ll look into it further.
Thanks for your help so far, everybody.
Thanks - went right over my head. Now that I’ve read it, 
I had (and still have) a pretty strong feeling that was the song you’re thinking of; it fits your description, I heard it often around 1987, and I haven’t heard it on the radio a single time since. If you’re having trouble finding a clip, email me and I can send you one (my email address is in my profile).
Oh, crap, sorry, I didn’t know my link to the clip of the song in post #24 didn’t work. Is there a format you know would work?
hoping I’m not crashing computers left and right 
Actually, your link does work (for me, anyway). I hadn’t tried it until just now; I just got an impression from Elendil’s Heir’s post that he/she hadn’t found a working link.
Yes, how could the SDopers forget? George Harrison released “I Got My Mind Set On You”, but it does not fit the description in the OP. - Jinx
Here’s a forgotten song from that time period, but I make no guarantees. Although you say it is not Phil Collins, Phil had a song only found on the CD of “No Jacket Required” at the end of the album that says: “…turn your head, don’t look back; set your sails for a new horizon, it gets better when you get there…” It’s called “We Said Hello Goodbye” but it may not meet your criteria.
- Jinx
Don’t overlook Pink Floyd for such a song. Pink Floyd put out the “Momentary Lapse of Reason” CD around this time of which “Learning to Fly” was the first single cut from this album. (This should not be confused with Tom Petty’s “Learning to Fly”.) There was another song cut from this entitled “On The Turning Away” which seems reminscient of older Pink Floyd songs of which many bordered on the seemingly insane. The third cut from this album “The Dogs of War” is lesser known. - Jinx
I’ve ordered The Smiths’s and Bourgeois Tagg’s appropriate albums through interlibrary loan, and should be able to find out when I get 'em. Cabbage, the more I think of it, the more I suspect that it’s the second. Thanks, everybody!
To clairfy for the pop fans of Billy Joel, his crazy period began with the Glass Houses album and continued into the Nylon Curtain album. But, both albums preceded the time period given in the OP. Still, the Nylon Curtain is an interesting artistic experiment on Billy Joel’s behalf to break away from mainstream pop music mold, IMHO… - Jinx
Have you tried www.allofmp3.com? It’s another site that sells mp3s either singly or as whole albums.
http://music.allofmp3.com/r2/The_Smiths/Strangeways__Here_We_Come/group_5512/album_1/mcatalog.shtml?albref=25 You may have to register to get there. That site lets you preview songs, just click the little speaker icon next to the song title (doesn’t show up until you mouse over the title)
George Harrison was a member of the Travelling Wilburys, who released an album about this time. Maybe someone who remembers that album better than I do would know if they had a song like this?
Except this one has a quicker tempo than what your description sounded like. I would definitely not call it slow and elegiac. Although it is primarily vocals and guitar (as described in the question).
Do you know any of the lyrics?
See posts #1 and 5.