You have been tricked, the website you linked is from a nasty Christian fundamentalist/Young Earth Creationist group. Radioactive decay is a physical constant (though one of probabilities), and as long as you have a reasonable number of C14 isotopes left in your sample (so anything more recent than, say, 8000-10.000 years old) you can get good certainty of your date.

We define time itself by radioactive decay through the use of atomic clocks.

That website is just another YEC site attempting to selectively reject the laws of physics.

Look at the source given to the specific claim that the rate of decay has been accelerated (I've added emphasis):

D. R. Humphreys, J. R. Baumgardner, S. A. Austin, and A. A., Snelling, "Helium diffusion rates support accelerated nuclear decay," in Proceedings of the Fifth International Conference on Creationism, R. Ivey, Ed., Creation Science Fellowship, Pittsburgh, PA, 2003. See also: Walt Brown, In the Beginning, 2001, p. 75, under "Constant Decay?"

I hope that give you an idea about the motivation behind the claim and the extent to which it should be taken at face value.

Another claim on the page your link is this, citing the same source:

We must also assume that the ratio of C-12 to C-14 in the atmosphere has remained constant throughout the unobservable past (so we can know what the ratio was at the time of the specimen's death)

This is not an assumption ever made in radiocarbon dating. The age is callibrated to account for the variability in the atmosphere's ^(14)C content, determined using e.g. dendrochonology (tree ring dating) and ice cores. On the following page the author attempts to discredit this method (thus blatantly contradicting the claim cited above), citing unwillingnes of scientists to disclose their data. This claim seems entirely unfounded, and cites e.g. this publication.

TL;DR: ^(14)C dating is a well established an entirely uncontroversial method. The page you've been looking at is extremely questionable, to put it very mildly.

Just a word as a geologist: there is nothing controversial about C^14 dating. It's a usefull tool. It has it's limits, which are well understood. Within these limits, there is plenty of usefull information which can be derived from it.

There are plenty of other methods for dating material: dendrochronology, speleothems, luminescence, palynology, varve stratigraphy, event markers, tephra, stratigraphy under various guises, K-Ar, U-Pb, etc ad nauseam...

Each has strengths and weaknesses, some are better adapted to a given time range (you want to date igneous rocks older than 10 MY? Forget C^14, use U-Pb), a type of setting (lake bottoms vs volcanic terranes) or material (growth rings and layers in mud vs wood vs stalagtites). Geochronology is a complex field unto itself, and C^14 is merely one of the tools at its disposal...

About the rate of decay, there is no reason to postulate it has varied across time. That is a physical constant. Beeing worried about whether physical constants vary in the "unobservable past" (not sure when that happened, observing the past is my bread and butter - my field of expertise mostly focusses on events 2.6 to 3.1 Ga ago. I observe plenty of that past... And I'd be an idiot to use C^14 in my rocks, they are way too old - I use U-Pb in zircons thank you very much) is akin to wondering if the laws of physics take a breather each time you blink. There is no indication that they do.

Edit: OHO! Seems you've been getting your info from creationists... That's your problem right there... Keep religion focussed on holy books and rituals, and go to the CRC press handbook of Chemistry and Physics for your science, and everything should work out all right in the end. (Please don't get me started on the sheer dingbat idiocy creationist ideology tries to foist away as "critical thinking".)

However, there is strong evidence which suggests that radioactive decay may have been greatly accelerated in the unobservable past

The guys at /r/askscience might know more, but a lot of astronomy/astrophysics researches if constants like radioactive decay and speed of light might have changed in the ~14 billion years the universe has existed, and most evidence hints towards constant constants