We present two new structure-preserving schemes for symmetric hyperbolic and thermodynamically compatible (SHTC) systems with involution constraints. First, we present a new staggered semi-implicit method that is able to preserve curl constraints exactly on the discrete level and which is also compatible with the stiff relaxation limit of the governing equations. In the second part of the talk we present a new thermodynamically compatible finite volume scheme that is exactly compatible with the overdetermined structure of the model at the semi-discrete level, making use of a discrete form of the continuous formalism introduced by Godunov in 1961 (semi-discrete Godunov formalism).