As newly independent countries after WWI, why didn't Baltic states have sovereignty dispute and war each other like other Eastern European countries?

They did.

There was the Latvian War of Independence lasting 5 December 1918 – 11 August 1920 (1 year, 8 months and 6 days) with participants from Latvian Army, merged from the: Latvian Independent Brigade, North Latvian Brigade, Estonia, Russia Lieven detachment, Poland, Lithuania, Supported by the Allied Powers, German Empire VI Reserve Corps, Baltische Landeswehr, Freikorps, West Russian Volunteer Army, Russian SFSR, Latvian SSR

There was the Estonian War of Independence lasting 28 November 1918 – 2 February 1920, (1 year, 2 months and 5 days). Participants were Estonia, Latvia, United Kingdom, Russia White Movement, Finnish, Danish, and Swedish volunteers Russian Soviet Federative Socialist, Republic Soviet Russia, Estonian Workers' Commune, Baltische Landeswehr.

There were the Lithuanian Wars of Independence:

also known as the Freedom Struggles (Lithuanian: Laisvės kovos), refer to three wars Lithuania fought defending its independence at the end of World War I: with Bolshevik forces (December 1918 – August 1919), Bermontians (June 1919 – December 1919), and Poland (August 1920 – November 1920). The wars delayed international recognition of independent Lithuania and the formation of civil institutions.

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Newly independent countries East of the Baltic Sea
After the First World War the term "Baltic states" came to refer to countries by the Baltic Sea that had gained independence from Russia in its aftermath. As such it included not only former Baltic governorates, but also Latgale, Lithuania and Finland.[15] As World War I came to a close, Lithuania declared independence and Latvia formed a provisional government. Estonia had already obtained autonomy from tsarist Russia in 1917, but was subsequently occupied by the German Empire; they fought an independence war against Soviet Russia and Baltic nobility before gaining true independence from 1920 to 1939. Latvia and Lithuanians followed a similar process, until the Latvian War of Independence and Lithuanian Wars of Independence were extinguished in 1920.

Update after question was edited:

The two borders now in question were disputed as well. That means that historic Livland (Livonia) was settled by Estonians and Latvians who did try to wrest as much control over territory. The German solution for this dispute would have been the United Baltic Duchy with multiethnic population. But not only were those ideas and those who had them unpopular, the idea came also too late, as the independence movements were already fighting against the Germans, the Russians, the Poles and each other.

From Wikipedia War of Latvian Independence:

Armies in Latvia during the Latvian Independence War on June 22, 1919 (stage 4) German army (around Riga) Latvian Bolshevik army (around Daugavpils) Latvian Ulmanis' army (trhee small pockets) Estonian army (around Valka)

Note the weasel words on Wikipedia. While it does say that the Estonians where there to help a Latvian government, it does not say that the Estonians also held a claim to territory there, hoping for recognition of these claims in return for "helping".

The British solution had something else on offer: Stephen Tallents helped to avert open warfare after independence, and he was instrumental in a treaty that partitioned for example Valka from Valga.

The Latvia–Lithuania border:

The border started to exist in 1918, when independence was declared by both states. There were some disputes on the exact location of the border but these were settled after a few years. These disputes related to places where the old Russian province borders did not follow the language distribution, and to railways.

Then it might be instructive to define what "dispute over sovereignty" actually means. All Baltic states might as well be described as suffering various degrees of civil war during the time…

Now we might still wonder why those conflicts and disputes between Baltic states appear to be relatively small. Border conflicts tend to be instigated by right wing politicians, nationalists and "patriots". But the population was quite war-weary and considered their chances in winning in violent confrontations weak at best. The reasoning in favour of an independent state had also to contend the extremely weak position an independent Latvia and Estonia had on the international scene

Only right wing extremists and students could be incited:

Once the Bolshevik forces had been expelled from Estonian territory, elections were held for an Estonian Constituent Assembly in April 1919. The results showed that the key issue for the majority of the electorate was land reform. The largest number of seats were won by the Social Democrats and Labour, parties who favoured swift and radical land reform – the immediate liquidation of the estates through division of the land. Without a just redistribution of land it was less likely that the common soldier at the front would have been motivated to risk his life for the Republic. (p10)

While the establishment of an independent Estonia was made possible by the profound changes in the international balance of power caused by the simultaneous collapse of Russia and Germany, Estonians were keenly aware that no state had been in favour of the formation of an independent Estonia. Thus they saw their own efforts and resolve as having been decisive. (p12)

Andres Kasekamp: "The Radical Right in Interwar Estonia", MacMillan: Basingstoke, London, 2000.

Source: Stephen Tallents: "Man and Boy", Faber&Faber: London, 1943, p 371. "The Latvian-Estonian frontier."