You're right, CFHTTP is the way to go. Here's a version of your call above translated to a CFHTTP call:
<cfhttp url="http://some.server/path" method="POST" result="resultName>
<cfhttpparam type="cookie" name="EASW_KEY" value="#EASW_VALUE#" />
<cfhttpparam type="cookie" name="EASF_SESS" value="#EASF_SESS_VALUE#" />
<cfhttpparam type="header" name="Content-Type" value="application/json" />
<cfhttpparam type="header" name="x-http-method-override" value="GET" />
<cfhttpparam type="body" value="#myJSONStringVariable#" />
</cfhttp>
<cfdump var="#resultName# />
The cfhttp tag is documented here: http://livedocs.adobe.com/coldfusion/8/htmldocs/help.html?content=Tags_g-h_09.html