  This is the first part of a series of two papers aiming to construct a
categorification of the braiding on tensor products of Verma modules, and in
particular of the Lawrence--Krammer--Bigelow representations. \\ In this part,
we categorify all tensor products of Verma modules and integrable modules for
quantum $\mathfrak{sl_2}$. The categorification is given by derived categories
of dg versions of KLRW algebras which generalize both the tensor product
algebras of Webster, and the dg-algebras used by Lacabanne, the second author
and Vaz. We compute a basis for these dgKLRW algebras by using rewriting
methods modulo braid-like isotopy, which we develop in an Appendix.
