Storage forum: WD My Passport Essential says Partial Backup

It's either Essential or Essential SE, as the paper that came with it says both. This is something that may be related to my old laptop's malfunctions, though I've found several topics in WD's support forums, all unresolved from last year or earlier, with the same problem. The laptop is an HP Pavilion dv6853cl with Windows Vista.

At the end of last month, when my old laptop went through a series of freezings, crashings, and shutdowns - and a result (from a test done at startup) that told me to replace the hard drive - I realized I might need a new computer. I plugged in the WD hard drive, which I hadn't done in a bit (no more than a month at most), and it said Partial Backup Accomplished. Previously it had said the backup was complete, but on the day of the shutdowns, the Other and Videos folders wasn't complete. I tried to make it keep backing up, but it was making no progress even though it was giving me the button for stopping the backup if I wanted. I would stop it, then start it, and still no progress.

Unfortunately I can't remember whether the problem showed up after the computer shut down while the drive was plugged in, or before that. In any case, later in the week I was cautious and didn't leave the drive plugged in for long, but now the problem had extended to other folders, like the Music folder. The only sections that were fully backed up were Documents and Pictures, and I've since stopped plugging it into my old computer - when it turns on and stays on - for fear that it will happen to that too. There wasn't any Music added since my last complete backup, unless my memory went really bad, so I don't think it was just backing up new data.

I can't see what files were saved either. That is, when I click on the drive in Computer - on another computer that I installed it on (on display at Costco, which an employee let me do, though he didn't help at all with diagnosing the problem) - I could see a folder of one user account, but not the ones from the account I used most often. My assumption, which I didn't even realize was an assumption until now, was that the files in the bar when I open WD Smartware were from the account I used most often.

Should I install the device in my brand new laptop and try retrieving the files?

Related, not trouble-shooting but pressing question - does this device store program settings? Not the programs themselves, but if I retrieve the backup on another computer, will, say, my IE browsing history be on the new computer when I open the browser?