If you truly don't have a privacy tab, you are apparently using a really obsolete version of IE. The "original" IE 5.0 that's with Win98 may not have had this tab(?) – I don't remember; but it should have been added if you've updated security patches within the past year or so, even if you've stayed with IE 5-series. (I don't have an older version to check, but my "perfect" recollection is that it was there on my last IE5 version. While my memory never fails, sometimes people change things that I "remember" when I'm not looking.) The IE4 that came with Win95 doesn't have a "Tools" tab, so I'll assume you're more current than that.
In Internet Explorer, click "Tools" at the top of the page. On the "rolldown" click "Internet Options." There should be 5 or 6 "tabs" at the top of the page, and "Privacy" should be one of them. (I'm not arguing with your statement that it's not there – just want to be sure we're talking about the same thing.)
You can check on what version of IE you're using by clicking on "Help" at the top of the main Internet Explorer page, and selecting "About Internet Explorer" at the bottom of the rolldown. You should see "IE 5.x SR1" or "IE 6.x SR1," where the "x" can be anything. If you don't show the SR1, or have something other than version 5 or 6, you are definitely "obsolete."
Just below the version, you should also see a list of updates, that should include at least 5 or 6 "Q" numbers. If they're not there, then you are "obsolete." Exactly which Qxxxxx numbers should be there depends on your IE version.
I'd strongly suggest getting the current version, since there have been a number of meaningful improvements - especially in "security" areas. You can get nearly all of the "fixes" for IE5, but they're getting so numerous that maintenance is a little cumbersome, and all of the significant older ones have been rolled into the current IE6.
You can get information about the current (6.0 SP1) version at Mickey'$ Support Center – Internet Explorer. Look here if you want more information before making changes.
In the "Support Menu" box at the left of the page, there is a "rolldown" for "Find & Download Software" that includes Internet Explorer. A click there will take you to the IE6 SP1 Download Page. You can go directly there if you just want to get the free download.
I currently have IE6 SR1 (+ updates) on machines running Win98, Win2000, and WinXP, and have had no problems with it on any of them, or with the conversion of any of them from IE5-series. IE is a "deeply embedded" part of Windows, so there's no way to guarantee a perfectly trouble-free upgrade; but my experience has been positive.
(DO NOT mistake "MSN Explorer" updates for "Internet Explorer" updates. "MSN Explorer" is a completely separate and "virtually unrelated" product designed to make sure you get the "maximum benefit" (to whom?) from MSN advertising.)
Since you can log in with each visit, but the cookie isn't "saved," it's most likely that you're getting a "session cookie" rather than a "permanent cookie," which is controlled by your browser. You may be able to find how to change this in your IE Help file by looking up "cookies."(?)