The usual theoretical description of drift-wave turbulence (considered to be one possible cause of anomalous transport in a plasma), e.g. the Hasegawa - Wakatani theory, makes use of various approximations, the effects of which are extremely difficult to assess. This concerns in particular the conservation laws for energy and momentum. The latter law is important in relation to charge separation and the resulting electric fields, which are possibly related to the L - H transition. Energy conservation is crucial to the stability behaviour; it will be discussed by means of an example. New collisional multi-species drift-fluid equations were derived by a new method which yields, in a transparent way, conservation of energy and total angular momentum and the law for energy dissipation. Both electrostatic and electromagnetic field variations are considered. The only restriction involved is the validity of the drift approximation; in particular, there are no assumptions restricting the geometry of the system. The method is based primarily on a Lagrangian for dissipationless fluids in the drift approximation with isotropic pressures. The dissipative terms are introduced by adding corresponding terms to the ideal equations of motion and of the pressures. The equations of motion, of course, no longer result from a Lagrangian via Hamilton's principle. However, their relation to the ideal equations also implies a relation to the ideal Lagrangian, which can be used to advantage. Instead of introducing heat conduction one can also assume isothermal behaviour, e.g. = constant. Assumptions of this kind are often made in the literature. The new method of introducing dissipation is not restricted to the present kind of theory; it can equally well be applied to theories such as multi-fluid theories without using the drift approximation of the present paper. Linear instability is investigated by means of energy considerations and the implications of taking ohmic resistivity into account are discussed. A feature of the results is that for purely electrostatic perturbations the second spatial derivative of the density profile plays a role, in contrast to the usual approximations. For a class of systems with , it is shown that linear instability can only occur when the resistivity is sufficiently large, while the Hasegawa - Wakatani theory predicts instability for arbitrarily small non-vanishing resistivity. It is shown that, for essentially electrostatic instabilities, magnetic perturbations in resistive systems may not be negligible even for . An example which will be treated in a future paper indicates in addition that, in systems with vanishing ion temperature, electron temperature profiles should strongly influence the stability via resistive effects. This is in addition to effects leading to -modes. It also demonstrates that, in general, it is not possible to perform an expansion with respect to the resistivity near . The new formalism is interesting not only from a theoretical point of view, but also, in particular, as a useful tool for numerical calculations.