We are using ORING make Ethernet switch model no RGS-7244GP-E_US
Integrated Realtek RTL8111DL Gigabit Ethernet controller
Add. Realtek RTL8168D/8111D PCI-E Gigabit Ethernet Card
Hope this will help answering my querry.
As far as i could see those cards aren't compatible with Teaming. The cheapest card that i know that you can team up is Intel EXPI9301CT or EXPI9301CTBLK in bulk version. Apparently your switch supports teaming 802.3ad ( Adaptive or Simetric ). It has a CLI and WEB interfaces but be careful. Network vendor usually say that their gear does a lot of stuff, but most times doesn't work or it is plain proprietary. On those cards the Teaming is driver mounted. Just install the driver, configure the teaming and you are good to go.
Fail over configurations don't need any type of switch. Remember, assembling a NIC Teaming, depending of the type that you want you are juggling with the fabric of L3/L2 mechanics. I mean, in your case, if there is a problem, just blame the switch, or recheck the configurations.