If Earth were to suddenly stop orbiting the Sun and instead start orbiting around it, the consequences would be catastrophic, but the timeframe for the Sun to freeze would be extremely long, on the order of billions of years.
The Sun's energy is generated through nuclear fusion in its core, where hydrogen atoms combine to form helium, releasing a tremendous amount of energy in the process. This fusion process sustains the Sun's high temperatures and prevents it from freezing.
If Earth were to stop orbiting and start falling directly into the Sun, the Sun's gravitational pull would rapidly accelerate Earth towards it. As Earth gets closer to the Sun, it would experience intense tidal forces, extreme heating, and eventually be torn apart. This process, known as "tidal disruption," would result in the complete destruction of Earth.
However, even with the Earth's mass added to the Sun, the freezing of the Sun would not occur for an incredibly long time. The Sun has enough hydrogen fuel to continue undergoing fusion for about another 5 billion years. This phase of the Sun's life is known as the main sequence. Eventually, the Sun will exhaust its hydrogen fuel and enter a different phase, where it will expand into a red giant and then shed its outer layers, becoming a white dwarf.
So, while the scenario you described would be catastrophic for Earth, it would not significantly alter the Sun's fate in terms of freezing. The Sun's freezing is a natural process that will occur billions of years in the future, unrelated to Earth's hypothetical change in orbit.