I think the Battle of Leyte Gulf is proof that the IJN was still a dangerous force after Saipan. Japanese aircraft carriers were certainly done after The Battle of The Philippine Sea as evidenced by their use as a decoy force in the Battle of Leyte Gulf. However, the remaining force of the Japanese Navy was considerable and quite dangerous.

The Japanese didn't manage to achieve any of their goals in the Battle of Leyte Gulf but the outcome of the battle wasn't guaranteed before the battle started. US forces were very lucky in the Battle. If Taffy 3 had not managed to turn Center Force around it likely would have made it to the landing area and possibly destroyed the landing fleet. Oldendorf's older battleships and cruisers would likely have defeated Kurita eventually but it is possible that Center Force could have ended the landings and postponed the capture of the Philippines for several months.

My vote is that Leyte Gulf represented the last offensive capability of Japan. Japan's surface fleet was largely destroyed and the capture of the Philippines cut Japan's oil supply line. I think Saipan and the Battle of the Philippine Sea were the beginning of the end of Japan's offensive threat and the end of the end was Leyte Gulf in October of 1944.

Did the Japanese pose any threat to the US after about July 1944?

It depends on how you define threat. Japan had the I-400 class submarine which were pulled from a mission to attack the Panama canal almost at the last minute to attack a major allied fleet anchorage when Japan surrendered. They had the Fu-Go balloons that were hitting the Northwestern Americas. There was a chance that the I-400s could have attacked the mainland, or that a Fu-Go might have hit something worth anything, but it wasn't really a big, perceived threat. The Seiran, which the I-400 carried, wasn't even known to the Allies until the war ended, so they wouldn't have taken its speed and range into account when defending points from submarine launched aircraft.

The bigger threat was to US forces, and as the Kamikazes show, they were still a threat to US forces even at the end of the war.

If your definition of "threat" is with regards to the mainland and even Hawaii, then Japan wasn't much of a danger after the Battle of Midway. But if you mean serious damage to US forces, including forward bases and ships, then they were a credible threat until the end of the war.