The majority of the research studies on Flex-Grid over multi-core fiber (Flex-Grid/MCF) networks are built on the assumption of fully non-blocking ROADMs (FNB-ROADMs), able to switch any portion of the spectrum from any input core of any input fiber to any output core of any output fiber. Such flexibility comes at an enormous extra hardware cost. In this paper, we explore the trade-off of using ROADMs that impose the so-called core continuity constraint (CCC). Namely, a CCC-ROADM can switch spectrum from a core on an input fiber to a chosen output fiber, but cannot choose the specific output core. For instance, if all fibers have the same number of cores, the i-th core in the input fibers can be just switched to the i-th core in the output fibers. To evaluate the performance vs. cost trade-off of using CCC-ROADMs, we present two Integer Linear Programming (ILP) formulations for optimally allocating incoming demands in Flex-Grid/MCF networks, where the CCC constraint is imposed or not, respectively. A set of results are extracted applying both schemes in two different backbone networks. Transmission reach estimations are conducted accounting for the fiber's linear and non-linear effects, as well as the inter-core crosstalk (ICXT) impairment introduced by laboratory MCF prototypes of 7, 12 and 19 cores. Our numerical evaluations show that the performance penalty of CCC is minimal, i.e., below 1% for 7 and 12-core MCF and up to 10% for 19-core MCF, while the cost reduction is large. In addition, results reveal that the ICXT effect can be significant when the number of cores per MCF is high, up to a point that equipping the network with 12-core MCFs can yield superior effective capacity than with 19-core MCFs.