Electronic – 60 LED chaser sequencer, two LEDs turn on at a time instead of one

4017 output is 'one hot', and won't source much over 1mA, so you can't use that directly.

If you must employ a 4017, then the simplest way to get to a base-5 chase sequence is to use diode logic to connect the 10 outputs to the bases of 5 NPN driver transistors. So output Q0 connects by 2 diodes to TR0 and TR1 bases. Output Q1 connects by 2 diodes to TR1 and TR2 bases, etc etc. Total bill of materials 20 diodes, 5 current limiting resistors on the bases, and five transistors. If you use a driver array like ULN2803, then you don't even need the base current limiting resistors. That array goes up to 50v, so you can put LEDs in series to reduce the current requirement.

Finally, use an oscillator to drive the clock pin. A variable speed clock varies the speed of the chase.

You're using a "glitch" to reset your counters. In other words, when the reset pulse starts, it immediately removes the conditions for its own creation, so it's only as wide as the propagation delay through one of the counters.

Clearly, one of those counters is faster than the other, so it resets successfully, while the other does not. This is why this is considered poor design practice, and why synchronous counting was invented — it only works under certain conditions.

The fix is to use the glitch to trigger a monostable timer (e.g., half of a 4098) that will guarantee the minimum reset pulse width for both counters. The reset won't occur until the timer is successfully triggered, by which time, it doesn't matter if the glitch goes away.

I see that you have removed R18's connection to ground, but I don't see any other provision to pull that node high. If that node is just floating, then you're just getting capacitive coupling and/or leakage current through the diodes for your reset pulse, compounding the problem.