Both HTTP GET and HTTP POSTS use TCP. If you are asking whether a POST also requires a 3-way TCP handshake (syn-synack-ack), it does just like any other TCP connection. The TCP handshake is required before any application protocol (such as HTTP) starts work.
FYI, your three-way handshake is incorrect; it should be "syn-synack-ack"
**ADD:**
If browser use QUIC (Quick UDP Internet Connections, pronounced quick. Proposed by Google) protocol for HTTP is possible to avoid 3-way TCP handshake. But AFAIK it supported in Chrome and Google.
Most software prefer HTTP/2 which still TCP but wit many features it use persistent connection then 3-way handshake done once for each server server.
If this protocols is used, 3-way hanshake can be avoided by any request, including GET.