Sulfurous acid tautomerisation
Most of what I would be saying now is stuff I already highlighted in another answer of mine to a related question.

$\ce{H2SO3}$ exists in two forms, one of which is equivalent to the structure you drew, the other is the structure you proposed. Taking the monoanions (because that was the focus of the other question and I am too lazy to redraw the picture) will give you these two structures:



(All horizontal bars are formal $-$ signs, never lone pairs.)

The crystal structure of $\ce{CsHSO3}$ was solved in 1980 by Johansson, Lindqvist and Vannerberg; they conclude that $\ce{CsHSO3}$ is not cesium hydrogen sulphite but rather cesium sulphonate.[1] See the other answer for details.

In cis-bisbipyridylbishydrogensulphito-&#x3ba;S-ruthenium(II) $\ce{[Ru(bpy)2(HSO3}\unicode{x2D}\unicode[Times]{x3BA}S\ce{)2]}$, a crystal structure solved by Allen, Jeter, Cordes and Durham, the sulphur atom directly coordinates the ruthenium centre which is only possible if it has a lone pair; thus we must be dealing with hydrogensulphite.[2]

Therefore, depending on the conditions both forms are possible and observed.

In case of the free acid, it is likely more stable in one of the two forms. I do not have analytical or thermodynamic data at hand to tell you which tautomer is preferred.



References:

[1]: L.-G. Johansson, O. Lindqvist, N.-G. Vannerberg, Acta Cryst. 1980, B36, 2523. DOI: 10.1107/S0567740880009351.
[2]: L. R. Allen, D. Y. Jeter, A. W. Cordes, B. Durham, Inorg. Chem. 1988, 27, 3880. DOI: 10.1021/ic00295a003.