Magnets to be used for the internal quadrupoles of an interdigital H-mode drift tube linear accelerator (IH-DTL) using KONUS beam dynamics should be both compact in size and high in focusing field gradient. Permanent magnets are an attractive solution, but then the ability to adjust the field strength is lost. We investigated two different solutions to this problem: the first using external adjustable electromagnets; the second using internal adjustable permanent magnets. The first method moves the variability out of the resonant cavity, using adjustable electromagnet quadrupole doublets before entry into the IH-DTL to compensate for the lack of internal variability. We carried out optimization simulations with custom code that ran many instances of the LORASR beam dynamics simulation software, using different values of field strength for the external doublets. By optimizing the magnet settings for different values of input current, we were able to compensate for the space-charge forces involved in accelerating a high-intensity continuous-wave (CW) deuteron beam. Second, we designed some novel adjustable permanent-magnet quadrupoles to be used inside the cavity, which combine the advantages of small cross-section and variable field gradient. This allows much more control over the beam, and even other ion species with differing charge-to-mass ratios can be accommodated within the same accelerator design. We developed two adjustable permanent-magnet designs: one with an electromagnetic component, and the other with two concentric moving rings of Halbach-array quadrupoles.