Transition-edge sensors (TESs) are the selected technology for future spaceborne x-ray observatories, such as Athena, Lynx, and HUBS. These missions demand thousands of pixels to be operated simultaneously with high energy-resolving power. To reach these demanding requirements, every aspect of the TES design has to be optimized. Here we present the experimental results of tests on different devices where the coupling between the x-ray absorber and the TES is varied. In particular, we look at the effects of the diameter of the coupling stems and the distance between the stems and the TES bilayer. Based on measurements of the ac complex impedance and noise, we observe a reduction in the excess noise as the spacing between the absorber stem and the bilayer is decreased. We identify the origin of this excess noise to be internal thermal fluctuation noise between the absorber stem and the bilayer. In addition, we see an impact of the coupling on the superconducting transition in the appearance of kinks. Our observations show that these unwanted structures in the transition shape can be avoided with careful design of the coupling geometry. The stem diameter appears to have a significant effect on the smoothness of the TES transition. This observation is still poorly understood, but is of great importance for both ac and dc biased TESs.