Czechia is a preferred short English name but Czech Republic is still the official full name of the country.
Short country name "Česko"/"Czechia" to be entered in UN databases
The name “Czechia” will not replace the full official name of the Czech Republic. It is simply the English version of the country’s short name (”Česko”) and as such it will be entered in the UN databases of country names. In fact, “Česko” has officially been the Czech Republic’s short name since 1993. It is part of a national standard established in that year by the Czech Office for Surveying, Mapping and Cadastre and approved (from the linguistic perspective) by the Czech Language Institute of the Academy of Sciences of the Czech Republic.
The full country name “Czech Republic” (“Česká republika”) will remain the country’s official political name. “Czechia” will be used only as the English version of the short country name “Česko”. It is up to each entity to decide whether to use the short version (“Česko/Czechia”) or long version (“Česká republika/Czech Republic”).
In some languages the Czech Republic’s short name is already well established e.g. in French - la Tchéquie, Spanish - Chequia, German - Tschechien.
(The emphasis is mine)
Source: Ministry of Foreign Affairs of the Czech Republic
Talking about Czechoslovakia CIA World Factbook states that
On 1 January 1993, the country underwent a nonviolent "velvet divorce" into its two national components, the Czech Republic and Slovakia. The Czech Republic joined NATO in 1999 and the European Union in 2004. The country changed its short-form name to Czechia in 2016
(The emphasis is mine)
Then the website of republic itself in a number of graphics lists the new short name, there are also articles on the website itself that discuss the name change. One of them says
It’s been a year since the word Czechia, a shorter version of the name Czech Republic, was entered in the UN database of geographical names of countries. To mark the occasion, the geographical department of the Faculty of Sciences in Prague organised a special conference, assessing how successful the process of adopting the name Czechia has been so far.
Source (This article is a must read)
Then this article on the official website details the process as well
The Czech Republic’s top officials are meeting to try and agree on a shortened name for the Czech Republic. The frontrunner in English appears to be the name Czechia and that could soon be popping up all over the place.
Then this one is very relevant to aliens so they know what to say
Earlier this month the Czech Foreign Ministry launched a five-minute video promoting the Czech Republic or should we say – Czechia. Apart from showcasing the country’s achievements the video is also meant to reinforce awareness of the recently approved short version of its official name.
Then we have this information from the UN
UN Statistics on Czech Republic
Czechoslovakia was an original Member of the United Nations from 24 October 1945. In a letter dated 10 December 1992, its Permanent Representative informed the Secretary-General that the Czech and Slovak Federal Republic would cease to exist on 31 December 1992 and that the Czech Republic and the Slovak Republic, as successor States, would apply for membership in the United Nations.
Following the receipt of their application, the Security Council, on 8 January 1993, recommended to the General Assembly that the Czech Republic and the Slovak Republic be both admitted to United Nations membership. Both the Czech Republic and the Slovak Republic were thus admitted on 19 January of that year as Member States.
On 17 May 2016 the Permanent Mission of the Czech Republic to the United Nations informed the UN that the short name to be used for the country is Czechia.
(The emphasis is mine)
TL;DR
We can use either of the names but their Government prefers/promotes that we use the shorter one in English.
