A covariant field-theory formulation of the recently developed dibaryon model of nuclear forces is given. The model involves an intermediate dibaryon dressed with π, σ, ρ, and other fields; the corresponding mesons can be in various orbitally excited states with respect to a six-quark bag; and scalar, pseudoscalar, vector, and axial-vector fields are taken into account for the dressed dibaryon. Particular attention is given to nucleon-nucleon interaction in even partial waves. A relativistically covariant nucleon-nucleon potential is derived. It is shown that a simple generalization of the single-channel model makes it possible to take into account coupling to other baryon channels of the NΔ, ΔΔ, NN*(1535), etc., types, this in turn leading to a description of the production of various mesons within this approach. The model also enables one to describe, in a natural way, 2π production in nucleon-nucleon collisions in the isoscalar and in the isovector channel and provides a new formulation of the theory of meson-exchange currents in the physics of electromagnetic interactions. The possible applications of the developed approach in the physics of hadron interactions and in nuclear physics on the whole are discussed.