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How about following the timer with a 40xx counter (not binary) with the reset on the second count. Count one activates relay #1, count two activates relay #2, then reset for the next pulse from the timer chip?
where two 4017s are put together to get a higher count than what you need. But, if you study the diagram and pull the data sheet on this IC you will get the jist of how it sequences and how it is forced to re-set at any predetermined point in the count.
There are other counters available worthy of consideration. Also, you may need a transistor switch to drive your relays since your IC output is rather weak.
A flip-flop driving a DPDT relay is another option worthy of your consideration.
I think a PIC is far too complicated for what you are trying to acomplish.
There is the flip-flop option, which I have shown below - the FF is half of a 4013 (dual D-type FF).
Alternatively, you may be able to use a "latching" relay. They're not very common, but are sometimes useful, if you can get hold of one. The relay is mechanically configured, so that it changes state when the coil is energised, and holds that state after power is removed from the coil. When you energise it again, it switches to the other state (and holds it). You can drive it directly from your signal (if it can supply the current) or via a transistor.
NB For the latching relay to operate correctly, the signal pulse needs to be fairly long - at least 100mS, maybe more - check the specs of the relay.