How come the cops didn't find the gun thrown in the shallow river? But they had his prints on the murder weapon

I'll start at the end.

Then how did they find his finger prints on the murder weapon?

They didn't, you've simply misremembered this part. The murder weapon was never recovered. Andy's fingerprints were found on a broken bottle and some bullets found near the scene of the crime. The revolver that the DA (shrewdly) shows at court is not the murder weapon but a substitute.

The prosecutor tells him that the police search the river for 3 days and didn't find any gun. How was this possible? The river was shallow, it was not a fast flowing river, the revolver was heavy

The Royal River is indeed a small river. But I feel this is plausible due to a few factors:

• The movie takes place in the 40s. The technological means and techniques for scouring and dredging that speed up a water search today weren't available back then;

• It was at night and Andy was still a little drunk, so he probably couldn't remember exactly where he threw the gun, which would make the search even harder;

• Chance. Simply put, these things happen. A police force isn't infallible, the river is still 40 miles long and Andy might've dropped the gun in a strong current that carried it away.

And ultimately, I'm not even sure it matters all that much. Without direct evidence, a murder weapon or eye witnesses, the DA put up a convincing argument: Andy had a motive, left fingerprints & footprints behind, admits he had a gun and bullets and his detached demeanor comes off as icy. If the gun had been found, the DA might've found a counter-argument (that Andy secretly had 2 guns, for example), because it's still Andy's word, and his story unfortunately doesn't ring true.