Why did Sirius stay there so long?

[Mindy, a.k.a. CLH]

No: HPFGUIDX 24038

I reread all of GoF today, but because it was the Jewish Sabbath I
couldn't write down any of my very burning questions and I forgot all of
them! It was really fun to reread it now that I know that Moody is really
Barty Crouch Jr.; now I understood for example why he hated Karkaroff so
much, and why he disliked Snape, etc.; not becaue he was an Auror, just
b/c he hates a Death Eater who walked free! The way he knew that
Longbottom was attached to the Cruciatus Curse and that Harry survived
Avada Kedavra -- all of it was clear now that I knew who he REALLY was.
However, I am still amazed that he managed to keep his wits about him,
after 10 or so y ears of living under the Imperius Curse.

But, I have one question about PoA. If Sirius was able to sneak out of
Azkaban disguised as a dog, why on earth did he wait thirteen years??? he
could've snuck out after 5 years and fled somehwere else. Or he could've
gone to some distant post office & written a letter to Crouch or s/one
else, explainng his innocence (since he never got a trial). Or whatever.
Why did he wait such a looooong time -- i Know the answer is that he
finally found out where Peter was hiding, but still, he could've left
Azkaban as a dog a lot earlier than he did.

Also, if he disappeared at the end of PoA, and he was 'at large' again,
why did the MoM let their guard down and let Harry go home, and remove
the Dementors? A very dangerous criminal is still at large! And they're
just leaving all their precautions behind. I dont' understand it.

Anotehr question. What is the difference between being an Animagi and
undergoing an animal Transfiguration, like the amazing bouncing ferret
Malfo or Krum the Shark? I don't see the difference. 

Thanks.