VLAN interfering with another VLAN on same switch

Hi r/networking,

I am trying to set up a switch (cisco sg300 series) to carry two different subnets (.10 and .20), with each on their own VLAN to avoid any contact between the two. Each subnet is being carried in on its own ethernet cable, to ports 11 and 12, respectively, and the subnets already exist independent of this switch and I can communicate with machines on either one that don't come through it.

I want to isolate the two of them, as they're being used for different purposes (communication/control on one, data transfer on the other) and having one be able to talk to the other has caused headaches in previous iterations.

Individually, each VLAN works as intended - I can communicate with machines on the .10 or .20 subnets individually, but if I have both plugged in at the same time the .20 subnet stops working and I can no longer ping any machines connected to it via the switch. If I remove the cable bringing in the .10 subnet or turn its port off, .20 becomes available.

What I want is a setup like the following (minimal) example:

VLAN	ports	use
1	13	(switch management)
2	11, 23	(11 comes in from external switch, 23 connects to a computer)
3	12, 24	(12 from ext switch, 24 connects to computer)

Is there any way to set up the configurations on the switch such that I can use both subnets simultaneously?

Thanks in advance.