This work uses a combination of neutron diffraction and bulk property measurements to establish the low temperature magnetic states in the dense metal-organic frameworks Ho(HCO2)3 and Er(HCO2)3; whose structures feature chains of face-sharing LnO9 polyhedra packed into a triangular lattice. Below 0.7 K Ho(HCO2)3 is found to adopt a state in which the magnetic moment on its ferromagnetic chains vary from neighbouring chains but the sum around each triangle is constant. Er(HCO2)3 is found to be the first lanthanide formate to adopt an ordered magnetic state with antiferromagnetic coupling within its chains near 50 mK. The potential to combine the ferromagnetic and antiferromagnetic coupling within chains associated with Ho and Er cations, respectively, in the same compound has also been explored via the series Ho1-xErx(HCO2)3. Ho0.5Er0.5(HCO2)3 remains paramagnetic to 0.4 K, suggesting it may be a starting point to search for a random spin chain paramagnet.