RSA is still secure. The largest RSA number factored to date is 829 bits. This is a long way away from the 2048 bit numbers commonly used in RSA today, and you can very easily go up to 4096 bits if you are concerned. RSA4096 will likely be secure for decades yet.

Shor's algorithm cannot yet break RSA, despite the clickbait news article which did the rounds earlier this year. The team in question managed to factor a 48 bit integer with a quantum computer, which is absolutely tiny even when compared to the 829 bit integer already factored by conventional computers, never mind the 2048 or 4096 bit integer needed to actually break RSA.

If RSA can be broken by quantum computers, then so can ECDSA.