How to solve Half Duplex/Full Duplex issue

The Ethernet adapter's LEDs usually do not mean "send" and "receive". Instead, one LED is a combined "activity" light for both sending & receiving, while the other LED indicates general status of the link, e.g. "green for 100Mbps / orange for 1Gbps" (or the other way around). So it's normal to see only one blinking, the other one should remain steady.

If you have Internet access – or even just if the computer was able to automatically get an IPv4 address – then the adapter is able to send as well as receive data.

"Half-duplex" means you can send and receive – just not at the same time. (That is, only one device can talk at a time, others must wait.) Very old Ethernet types worked this way, whereas modern switched Ethernet is full-duplex and allows both sides to talk at once. Wi-Fi is also half-duplex.

(If a connection is strictly one-way and can only send, or only receive, that's "simplex".)

Usually your devices should not be manually configured for either mode at all – that is, they should be allowed to auto-configure the Ethernet mode. Auto-negotiation is practically mandatory for Gigabit Ethernet, and it's not surprising that your router won't let you switch it off.

(The manufacturer might have omitted the option because in home networks it's very rarely necessary, and easy to cause problems. For example, if one side is expecting auto-negotiation, but the other is set to a manual speed/duplex mode, then the first device will give up and choose some default parameters which might not match what the other one was set to.)