The API is working for me. I would make sure you are using the latest CRM module (now 13.0.4.34), make sure the the API is enabled (Settings, CRM API Settings) and test from the pbx bash prompt:
[root@34693894 asterisk]# curl -H "Token: token_goes_here_xxxxxxxxxxxxxxxxxxxxxxxxx" http://localhost/admin/api/sangomacrm/rest/calls/list/both | python -m json.tool
% Total % Received % Xferd Average Speed Time Time Time Current
Dload Upload Total Spent Left Speed
100 62226 0 62226 0 0 622k 0 --:--:-- --:--:-- --:--:-- 626k
{
"data": [
{
"calldate": "1516841910",
"cnam": "",
"cnum": "",
"destination": "920xxxyyyy",
"direction": "outbound",
"exttocall": null,
"owner": "",
"recordingfile": "",
"source": "5002",
"type": "START",
"uuid": "1516841910.23",
"voicemailfile": null,
"zulu_type": null
},