The tightness of the layout probably stems from low resolution screens. In '96 the most typical display size was something around 640x480 or 800x600, not a lot of room to breathe there.
In the last month, I've had 2 people tell me (unsolicited of course) that my website looks like it was made 10 years ago.
I went to this 1996 campaign site and said with dismay, it looks like my website.
I recognized many of of the above historical attributes (table-based layout, centered narrow, body bgcolor attribute) in my own site.
I've got a full-time job as a UNIX system administrator, and a part-time business training sys admins. I don't have the website redesign merit badge or time to earn one. I'd pay for a redesign but I don't even know what to ask for except "a website that presents my company well". And I want people to stop staring at my website like there is something wrong with it.
Would appreciate any help you can offer, either by recommending a designer or a design, please.
If you don't mind looking a bit like other sites, I'd highly recommend using bootstrap as a basis for a modern-looking redesign. You can customize it to your will, or even buy some bootstrap-based layouts to work with for not much money: https://wrapbootstrap.com/
As someone who learned table-based layouts and who now designs using HTML5 and CSS3, rest assured that the "new" web is not that hard to learn as long as you understand the basics of css. And almost all your JavaScript skills will transfer. It's really as easy as picking up a book on HTML5 and reading it. Alternately, check out: http://diveintohtml5.info/
Unless your site is a differentiator for your business, I wouldn't go hog-wild on it. I'd start by looking at something like ThemeForest, where you can browse around, see if you find something you like, and buy it for not-a-lot-of-money.
