Were officers allowed to resign during the American Civil War?

Short answer yes.

McDowell took the brunt of public vituperation for the defeat at Bull Run, but Scott, who had helped plan the battle, also received criticism.[190] Lincoln replaced McDowell with McClellan, and the president began meeting with McClellan without Scott in attendance.[191] Frustrated with his diminished standing, Scott submitted his resignation in October 1861. Though Scott favored General Henry Halleck as his successor, Lincoln instead made McClellan the army's senior officer.[192]

https://en.wikipedia.org/wiki/Winfield_Scott#Lincoln_administration

McClellan was nominated by the Democrats to run against Abraham Lincoln in the 1864 U.S. presidential election. Following the example of Winfield Scott, he ran as a U.S. Army general still on active duty; he did not resign his commission until election day, November 8, 1864.

https://en.wikipedia.org/wiki/George_B._McClellan#1864_presidential_election

There were no other cavalry officers available with an earlier date of rank, and Dodge refused to be subordinated to someone who was junior to him in seniority. He resigned in June 1863 in protest, and briefly returned to service with the militia to suppress the New York City draft riots.

https://en.wikipedia.org/wiki/Charles_C._Dodge

These examples show that Generals could resign from the US military during the Civil War. People who are more expert should be able to provide examples of lower ranking officers resigning from the United States Army, or the United States Volunteers, or the militias of various loyal states.

Generals were not the only ones allowed to resign. In the movie They Died With Their Boots On (1941) southern cadets are allowed to resign from West Point.

I quote my answer at:

https://moviechat.org/tt0034277/They-Died-with-Their-Boots-On/58c702ac4e1cf308b933cf4d/Did-the-southern-cadets-walk-out

In real life not only were southern cadets allowed to resign, but southern born officers were allowed to resign if they had completed their required years of service - commissioned officers were required to serve for a few years.

High ranking officers who resigned and joined the Rebels included Colonel and Adjutant General Samuel Cooper, Colonel Albert Sidney Johnston, Brigadier General and Quartermaster General Joseph E. Johnston, Brigadier General David E. "Traitor" Twiggs, and of course Robert E. Lee.

Fortunately such leading southern born officers as Quartermaster General Montgomery C. Meigs, General Winfield Scott, and General George Thomas remained loyal.

After one battle General Custer greeted a captured Rebel general with a hug - they had been friends at West Point.

And this https://en.wikipedia.org/wiki/List_of_American_Civil_War_generals article says:

At the outbreak of the Civil War, 296 U.S. Army officers of various grades resigned. Of these, 239 joined the Confederate Army in 1861 and 31 joined after 1861. Of these Confederate officers from the U.S. Army, 184 were United States Military Academy graduates. The other active U.S. Army 809 officers, 640 of whom were West Point graduates, remained with the Union. Of the approximately 900 West Point graduates in civilian life at the beginning of the war, 114 returned to the Union Army and 99 joined the Confederate Army.[51]

If you look up a source that lists all the officers in a civil War regiment, you will see that some of them died, were killed, were discharged for disabilities, were missing in action, or were discharged with their units, and some of them resigned their commissions.

I suggest that you look at an online copy of Heitman Historical REgister and Ditioairy of the Unted States Army https://archive.org/details/historicalregis03heitgoog/page/n8/mode/1up

I thas several lists of officers of the regular amry and the US Volunteers in different conflicts. You will find examples of officers who resigned during the course of different wars.

Enlisted men were required to serve for the full terms of their enlistments.
In 1862 the Rebel Army forced all the soldiers who had volunteeered for one year in 1861 to remain in the army for the duration of the war.

But the United States did not act so tyrannically. In 1863-1864, when many Union soldiers who enlisted for three years in 1861 were approaching the ends of their enlistments, the US government went to great efforts to presuade them to reenlist voluntarily, instead of arbitarily decreeing that they had to serve until the end of the war like the Rebel government did. The majority of those soldiers did reenlist, but many thousands did not and were discharged honorably at the expiration of their terms.