Well I've sat on answering this one for a month (my first answer anyway) but since Belfast's sunshine comes in a liquid form today I've got some time to knock this together.

You're right, of course, Unionism didn't die out in the Free State after 1918 nor even when the Republic officially formed. Although it didn't transform into the fever of Loyalism which developed in the North it's easy to forget that some Unionist figures, obviously Edward Carson, were Dubliners themselves. A heads up that this answer is indebted to the wonderful collection of essays Unionism in Modern Ireland: New Perspectives On Politics and Power edited by Richard English and Graham Walker (but not wholly).

To a get a grasp on what Unionism is, and thereby, identify who supported it and how they were faring in the Free State and Republic it's critical to identify what Unionism is not: i.e. it is neither Loyalist nor is it Orangeism.

• Orangeism is the oldest of these isms, and the funnest word, and is wholly Protestant - it is an unquestioned belief in the role of the Loyal Orange Order in protecting and promoting the rights of the Protestants of Ireland (esp those who celebrated the Twelfth, the Boyne, and all that hoopla).
• Loyalism is the extremist variation of Unionism. A Loyalist swears to maintain connection to the British state by any means necessary: i.e. the paras. Loyalism is entirely fascinating and further discussions of it are a great "through the looking glass" material as it essentially turns back on itself through the use of betrayal myths and British debts (that's a question for another time)

Unionism, instead, developed as the opposition political thought to Home Rule. While it's strength in Ulster had more to do with religious identity and a feeling of otherness from the island, it was essentially a conservative political opinion. Amongst the wealthy in the 26 counties many had spent time abroad in England, either working or being educated, and so many remained loyal British subjects. They also saw Home Rule as a risk to the status quo and their wealth: Irish Catholics had made great advances to their rights since the famine and many feared that Home Rule would break their own economic grip on the country or even force them to give up their holdings. A number of large, still extant firms, were unionist such as Guinness, Beamish, Jameson (there's a theme here), Smithwick's, and even The Irish Times. That is to say, before the 1918 election Unionism was more simply a conservative opposition to Home Rule and hadn't become caught up in Orangeism or Loyalism that would later problematise it as a position.

Let's look at that Irish Election. Sinn Féin, as you've started, absolutely cleaned the floor with Carson's Irish Unionists and John Dillion at the helm of the Irish Parliamentary Party. Here's a table to quickly parse the results

Party	Votes	Seats
Sinn Féin	497 107	73
Irish Unionists	257 314	22
Irish Parliamentary Party	220 837	6
Labour Unionist	30 304	3
Belfast Labour	12 164	0
Independent Unionist	9 531	1
Independent Nationalist	8 183	0

I'm leaving off the indie and indie labour results here because they drop into the hundreds. Together Unionist votes equate to about 30% of all votes cast, and thousands of these votes were cast in what would become the Free State. Unionists across Ireland were of a sizeable number and were extant outside of traditionally Protestant Ulster. What changed the fortunate of that unionist minority in the South after the election of 1918, however, was the War of Independence and the Irish Civil War.

Peter Hart has identified that the common occurrence of Southern Unionists during this period of conflict was, as much as possible, abstention. Those loyal to the crown saw the violence around them and opted to remain aloof from it - they did not turn evidence (tout by Irish parlance) on the IRA. Hart's work interviewing Unionists found that most feared for their lives (or at the least they feared for the security of their property) and would rather make themselves politically unimportant than to take any active role. Fear was certainly the correct option; sectarian killings of Protestants spiralled into the hundreds in the period of 1920-21 (Hart, again). They saw that republicanism held the day and that, in the Free State at least, they were no longer of any real consequence.

Although even after the formation of the Free State (and then we can debate when it truly became independent) Unionists lingered on in the South. Patrick Maume considers the conflicted lives of three writers, who he describes as "Cultural Unionists", in the Free State. By the 1930s all three had begun to drift away from Ireland but found themselves unmoored in England: each felt Irish but not republican Irish, and English society certainly never accepted them. Maume, through these examples, touches on common experiences for Southern Unionists. Many drifted away, although some remained in Dublin and Cork. Those who shifted to Ulster were willing to accept the changing, cultural landscape of the North: religious - Protestant - fundamentalism continued to grow and Belfast was a dirty, industrial city (literally, Belfast didn't even develop a working sewer system until exceptionally late by UK standards: Dublin had 55km of sewer in 1849, Glasgow built 50 miles of it by 1875, Belfast first opened some in 1888 but it wasn't until the 1900s that it developed a sewer system). Others chose to further emigrate, their children absorbed into British society and similar societies in Canada, New Zealand, and Australia. And still, some stayed, grew silent, remained in their stately homes and positions of corporate authority (some of these individuals would find themselves targeted by the Provos during The Troubles, but that's for another question).

Peter Hart, "The Protestant Experience of Revolution in Southern Ireland", Patrick Maume, "Ulstermen of Letters", and Ian McBride, "Ulster and the British Problem", all have deeply fascinating essays, titles given, on this topic in Unionism in Modern Ireland: New Perspectives On Politics and Power ed Richard English and Graham Walker.