It's not exactly your situation, but I can say that for the past several years I have contributed to both an FSA and an HSA together. The FSA was configured as an "HDHP compatible" account, and was only valid for dental and optical until I met my deductible. Upon proof of meeting that deductible (this was not done for me, there was a form and I literally had to fax over all my insurance statements) the FSA could be opened up for all medical spending, like normal.
This arrangement was complex, but allowed me to max my HSA and keep it invested while still getting some relief on medical costs. Besides the limit on types of spending, there is also a lower limit on contributions to the FSA--not sure if that is because of the HDHP arrangement, or some other health law change.
So, I think they are blowing smoke about not contributing to both, but there may be some paperwork / configuration to the FSA to continue to hold it. Or perhaps your custodian does not have this option. Was it the custodian who told you that you could not contribute to both, or the plan admin or HR or someone? I would make sure to get as close to the source as possible: the custodian who holds your FSA.