Why do I get continuity between the whip portion...

Typically the capacitance of the cable will increase the resistance between the center conductor and shield of the cable, and the antenna will generally have a coupling capacitor prior to the connection to the cable. But between the whip portion and the center, there was continuity and there was continuity also between the whip and shield portion. The DC resistance of any antenna is not the same as the total impedance of that antenna. All DMM's measure only the DC resistance and not the total impedance of any item.
For example, measure the resistance of any "8 Ohm" antenna with a DMM. The resulting measurement will vary, generally as between 3-6 Ohm. This is only showing the "DC" component of the total impedance.
The antenna will radiate the signal properly and not burnout the transceiver as long as the SWR (Standing Wave Ratio) remains between 1.0 and 2.5. The antenna can work as long as the impedance at the desired frequency remains 50 W and the SWR remains between 1.0 and 2.5. Once again, the measurement is only accurate for DC (0 Hz). Antennas are tuned for operation within certain band limits. The impedance can only be tested at that frequency or band of frequencies.
There are other components in the manufacture of an antenna. Internally, the antenna will have a combination of components to help tune the antenna to the proper wavelength and power rating. Some antennas utilize inductors and capacitors, others use resistors and capacitors, still others may combine one or all of these in the design of the antenna. These "tuning circuits" can be a combination of LC, LR, RC, or even LRC (L = inductor, R = resistor, C = capacitor). The most common is an inductor or coil, sometimes referred to as a "loading" or "base" coil. The resistance of almost any coil will measure less than 10 W at DC or 0 Hz.
These concepts can be studied further via reference books such as the ARRL handbook and/or ARRL Antenna book.