First of all, my game settings are as follows:

Vs. 9 Races - 8 Minor Races - Tough Difficulty - Gigantic Galaxy
All Set At "Abundant", with "Loose Clusters" or "Scattered"
I play as Terrans, heavily focused on Tech & Diplomacy

Game runs fine for a few hundred turns. However, about 250+ turns in
my game becomes a little slow, and gradually progresses to become agonizingly slow.
I expect some of that. As races expand and things get more complex, clicking
TURN expectedly becomes a longer process, and that's fine.

The issue I'm having, occurs OUTSIDE of a period when my PC is "Running a turn"
My problem is, clicking on objects (ships, starbases, planets, etc.) becomes a
sit and wait process. I click something, and the game pauses, and seems to "Hang".
At 200 - 250 turns, the game's "wait-for-me-to-think-about-it" time is only about 3
seconds before the screen goes into auto-focus on the object (Or calls up the appropriate menu) But at, say, 700+ turns, this wait time is more like 4 minutes per-clicked-object!

I use the Blind Exploration option, and the more of the map I've
uncovered, and the more everyone colonizes planets and builds starbases, the longer my click-and-wait time becomes. Any clue on why this is happening? Is there anything I can do to fix this?

My PC pushes a lot of newer games much faster and harder, so I doubt it's my config. Not sure though. In any case, here are my core system Specs:

AMD Athlon XP 2800+ ATI Radeon All-In-Wonder 9800 Pro (Single Montior Setup)
1 GB (512 x 2) DDR (x4) W.D. 750 GB HDD (3 TB Total Non-RAID Array)
Audigy 2 Plat. Audio
*Nothing is OC'd - 13 Internal Fans Total - 14-Bay Case - 750 Watt Antec PSU