You need below 2866 m/s of orbital velocity at 1 AU to crash into the Sun.
You technically don't need to slow down exactly to 0 m/s relative to the Sun in order to crash into it. Let's calculate the approximate velocity required to graze the "surface" of the Sun. This is an excellent answer on how to calculate apoapsis and periapsis of an orbit.
So first, the Earth is about 150,000,000 km from the centre of the Sun. We want to obtain a perihelion of 700,000 km from the centre of the Sun (radius of the Sun is about 697,000 km, so that's about 3,000 km above the "surface").
So let's work backwards.
To calculate eccentricity, use:
$$e=\frac{r_a-r_p}{r_a+r_p}$$ which is $$e=\frac{1.5 \times 10^{11}-7 \times10^8}{1.5 \times 10^{11}+7 \times10^8}$$
therefore, $e = 0.99071$. Now let's find what velocity we need at apoapsis (starting point) to have a periapsis of 700,000 km. Let's work backwards.
$$a = \frac{r_p}{1-|e|}$$ which is $$a = \frac{7 \times 10^8}{1-0.99701}$$ and therfore, $$a=7.535 \times 10^{10}\space m$$
Calculate orbital specific energy (we need to use the Sun's GM which is $1.327\times 10^{20}$):
$$E=\frac{-GM}{2a}$$
so,
$$E=\frac{-1.327 \times 10^{20}}{2 \times (7.535 \times 10^{10})}$$
and therefore, $E = -880557398.8$. Now we just calculate velocity at 150 million km.
$$V=\sqrt{2(E+\frac{GM}{r})}$$
substitute values (remember, $r$ is 150 million km).
$$V=\sqrt{2\bigg(-880557398.8+\frac{1.327 \times 10^{20}}{1.5 \times 10^{11}}\bigg)}$$
and $V = 2866.8$ $m/s$.
We can conclude that we need about 2867 m/s of velocity at the distance of 150 million km to obtain a periapsis of 700,000 km which is just above the surface of the Sun. Meaning you need a $\Delta V$ of $-26.914$ $km/s$ because Earth's velocity is about 29 km/s. Since 26 km/s of delta v is A LOT, what most spacecraft do is go to one of the outer planets (like Jupiter) and use a gravity assist to decelerate. Orbital velocity decreases with distance.
And Earth would lose its orbital energy and spiral and crash into the Sun but that would take billions of years. Satellites take many years to de-orbit Earth because of the atmosphere and the Sun's activity. But before Earth even loses its orbital energy, the Sun would expand into a Red Giant and possibly swallow Earth.