Initialize swarm mode:
root@ip-172-31-44-207:/home/ubuntu# docker swarm init --advertise-addr 172.31.44.207
Swarm initialized: current node (4mj61o
The simpliest command I found to docker exec into a swarm node (with a swarm manager at $SWARM_MANAGER_HOST) running the service $SERVICE_NAME (for example mystack_myservice) is the following:
SERVICE_JSON=$(ssh $SWARM_MANAGER_HOST "docker service ps $SERVICE_NAME --no-trunc --format '{{ json . }}' -f desired-state=running")
ssh -t $(echo $SERVICE_JSON | jq -r '.Node') "docker exec -it $(echo $SERVICE_JSON | jq -r '.Name').$(echo $SERVICE_JSON | jq -r '.ID') bash"
This asserts that you have ssh access to $SWARM_MANAGER_HOST as well as the swarm node currently running the service task.
This also asserts that you have jq installed (apt install jq), but if you can't or don't want to install it and you have python installed you can create the following alias (based on this answer):
alias jq="python3 -c 'import sys, json; print(json.load(sys.stdin)[sys.argv[2].partition(\".\")[-1]])'"