CRJ700 Captain reported there was a strange odor that appeared and dissipated shortly after takeoff and during the descent to the destination airport; the flight crew began to experience physiological symptoms. The flight attendants and passengers were unaffected by the fume event.

Narrative

Upon preparing for departure; the right pack was not cooling properly; with the cabin temperature at 26 degrees C and not improving beyond 27 degrees C. This was unusual given the cooler weather; and the heat appeared to be increasing; but no caution messages were present. Although the pack abnormality seemed strange; it may or may not have been a factor in the event that followed. FAs confirmed it was warm but acceptable.Shortly after takeoff; both the FO and I noticed a strange odor in the flight deck that lasted about 10 seconds before dissipating. I checked with the FAs; and the forward FA also reported the smell; but the aft FA did not notice it. We checked each individual system for abnormalities; and there were no abnormal indications or caution messages on the EICAS. After discussing it; we elected to continue the flight but remained vigilant.As we descended to ZZZ; I began feeling faint; experienced tingling; had a rapid heart rate; and was having trouble breathing. I informed the FO and put on my oxygen mask. The FO initially said she was fine; but after seeing my symptoms; they decided to put on their mask as well a little after me. Shortly after doing so; she reported feeling faint as well. At this point; I identified the situation as a possible fume event. I'd like to point out that there was no smell associated with this phase of flight.We immediately requested a lower altitude from ATC and [requested priority handling]. My main priority was to remain conscious and get the plane on the ground safely. I checked in with the FAs; who reported that they and the passengers were feeling fine. Passenger oxygen masks were not deployed. I briefed the FAs on our ETA; approximately 20 minutes; reassured them that it would be a normal landing with no need for a brace signal; and instructed them that fire trucks were being rolled as a precaution.The decision was made not to notify the passengers as it was determined; after speaking with the FAs; that both the crew and passengers were fine; we were landing at the original destination; and it would be a normal landing. There was no immediate threat to the passengers; and with the focus on staying conscious and safely landing the aircraft; there was little time to make a PA announcement.We performed the QRC/QRH for smoke/fire/fumes and completed the necessary checklists while being vectored for a straight-in approach to Runway XX. We landed without incident; vacated the runway slowly; and determined that fire truck assistance was not necessary. We taxied to the gate while still wearing oxygen masks and kept the flight deck door closed until the aircraft was fully parked. After the firefighters arrived; we opened the flight deck door.The firefighters conducted checks and reported no abnormal oxygen levels. I confirmed with the FAs that both they and the passengers were okay. After deplaning; the FO and I provided statements to Dispatch and the chief pilots. EMTs took our vitals; pulse oxygen and blood pressure; and although we declined hospital transport; we continued to feel unwell after they had left.We contacted Maintenance Control; and Maintenance began checking the aircraft to determine the cause of the potential fume event.A few hours later; still experiencing symptoms; the FO and I went to the emergency room. We described our symptoms to the attending doctor; who mentioned that the symptoms were consistent with carbon monoxide poisoning. Although our bloodwork came back clear by the time it was tested; around XF:00; the doctor explained that with 100% supplemental oxygen; the body eliminates approximately 50% of carbon monoxide every 90 minutes. Since our symptoms started around XA:00 and our bloodwork was drawn at around XF:00; the carbon monoxide would have already metabolized by the time of testing. The doctor noted that while the paperwork stated no evidence of carbon monoxide poisoning;" this could not be ruled out due to the delayed testing and our consistent symptoms.Maintenance Control and Maintenance are investigating the cause of the potential fume event. All passengers and crew remained safe throughout the incident.Suggestion: Carbon monoxide detectors in planes for possible fumes and carbon monoxide training for pilots; at least knowing what symptoms to be aware of. This could help speed up the decision of using oxygen masks."