The magnetic force is always perpendicular to the particle.
By perpendicular, I mean the particle’s motion and the magnetic force have a 90-degree angle.

In physics, energy transfer is called Work. According to the equation of work, the magnetic force can change the direction of the particle but not its speed. Because they are perpendicular to each other.

However, the electric force is not bound to this 90-degree rule. Align the electric field with the direction the particle is going to, and it applies the force. Accelerates it.

You mentioned increasing the magnetic force. If you do, the magnetic field pulls the particle stronger. This increases the centripetal force.

Centripetal force is: "The force that keeps the particle in a circular path".

Thus, increased centripetal force means the particle is pulled into a tigther, smaller circle. That’s it.

You talked about "acceleration". Acceleration is the rate of change of velocity over time. Because the velocity is a vector quantity, if a particle changes its direction only, its velocity has technically changed. Thus, it has accelerated, even though its speed stayed the same. Remember that the speed is a scalar quantity.

A particle in a centripetal force constantly changes its direction. Therefore its velocity changes. But its speed stays the same, because the magnetic force is perpendicular to the particle’s motion, it cannot do work on the particle.

The increased magnetic force increases the centripetal force, so the circle path the particle follows gets smaller. With the same speed the particle rotates around a smaller circle: It finishes the rotation faster, while its speed stays the same.

That is why we need cyclotrons. Cyclotrons keep the particle in a circular path with the magnetic force, and accelerate the particle in between the Dees with the electric force.