Poor quality suspended ceiling installation
Yes, there are several alternate ceilings that could be installed: 1) suspended ceiling with decorative panels, 2) gypsum board on suspended metal runners, 3) gypsum board applied directly to floor supports,

First, if the existing tiles have water stains (which turn brown) and sag, it could be due to a leak. You’ll need to find and fix the leak before installing a new ceiling. 

1) There are many decorative suspended ceiling options. Here’s some options:  https://www.decorativeceilingtiles.net/suspended-drop-ceiling-tiles/

Or:  https://www.proceilingtiles.com/

2) Because your existing ceiling is suspended and you like a standard gypsum board ceiling (to match the rest of your house), you may need to suspended it on metal runners. Like this:
http://web.nwcb.org/CWT/External/WCPages/WCWebContent/WebContentPage.aspx?ContentID=98

3) The most common (and probably cheapest) is gypsum board applied directly to the bottom of the floor supports. However, if you have heating ducts, electrical wiring, etc. under the floor supports, then moving all that will be expensive.