Next generation heavy ion therapy and research facilities require efficient accelerating structures. Particularly, at low beam energies, right after the standard scheme of the ion source, low-energy beam transfer, and radio-frequency quadrupole (RFQ), several options for accelerating structures are available including the classic drift-tube linac (DTL), the interdigital H-mode DTL (IH-DTL), and superconducting quarter-wave resonators. These structures need to integrate the beam acceleration with the focusing channel, nowadays typically provided by permanent-magnet quadrupoles (PMQs). The frequency of operation needs to be in line with that of the RFQ structure, and it has been chosen at 750 MHz for practical considerations for the Next Ion Medical Machine Study (NIMMS) that is the application focus of this manuscript. While classic DTL structures at low ion beam energies do not provide enough space for PMQs at that frequency within a single $\ensuremath{\beta}\ensuremath{\lambda}$ period, IH-DTL structures do not provide the regular focusing channel with consequences on the beam quality. For these reasons, quasi-Alvarez drift-tube linac (QA-DTL) structures are reevaluated in this manuscript as they might fill this gap. They have not received much attention in the literature so far and therefore their design is described in detail. The design procedure presented here may serve as a blueprint for DTL design in general. In addition to the overall rf design, axial field stabilization with a new technique and multiphysics studies of the rf structure are described. A cost estimation completes the NIMMS QA-DTL study.