Need some car help

I am having some trouble with my 02 Buick century recently. The acceleration from a stop is very sluggish and gear changes come slowly, although they still seem to shift smoothly. The first thing I did was check the fluid levels and I can find nothing wrong there. The trans fluid was last changed 3 years ago, but the level, color, and smell are all normal so surely that couldn’t be the culprit?
Unfortunately, that about exhausts my diagnotic skill. I’d take the car to the shop if I could afford to, but I can’t so I hope with some kind help from you fine folks I might be able to understand the problem at least, even if I am unable to rectify it.

When you say the gear changes come slowly, do you mean the transmission is shifting late and revving up really high or just that the car is taking forever to accelerate to the normal shift points? I assume there’s no Check Engine Light or anything like that?

Quick, simple WAG… Your parking brake isn’t engaged, is it?

There’s no tachometer, but when I give it gas it sounds like the revs go up yet there’s no extra power. It takes ages for it to accelerate to where it changes to a higher gear.

No. I have done that before a few times when I first learned to drive, but not this time. I live in a very flat coastal plain so I never need to engage the parking brake in any case.

Possibly an exhaust system restriction.

Manually put the transmission in the lowest gear. Does it accelerate better then? When you get to about 20 mph and shift to the next gear, does it shift quickly?

If it accelerates the same and doesn’t seem to shift, the transmission could be in limp home mode.

If it accelerates the same but does seem to shift, then it’s probably not the transmission.

If if does accelerate better, then it might be something wrong with the transmission not automatically shifting to first gear.

How about the CEL/SES light on the dash? Is it on? It might be a little red or yellow light that looks like an icon of an engine. There could also be a transmission light on the dash or an OD for overdrive light on the dash. Those two might indicate a transmission issue.