This seems to work, Ubuntu is hoarding port 53 though even though local services are shut off.

Config/DNS/Dockerfile

@@ -6,15 +6,16 @@
 
 FROM debian
 
+# Remove Existing Config
+RUN rm -rfv /etc/{hosts,resolv.conf,dnsmasq.conf}
+
 # Install Dependencies
 RUN apt update && apt install -y dnsmasq
 
-# Remove Existing Config
-RUN systemctl stop dnsmasq
-RUN rm -rfv /etc/{hosts,resolv.conf,dnsmasq.conf}
-
 # Copy Configuration Files
-COPY ./config/{hosts,resolv.conf,dnsmasq.conf} /etc/
+COPY ./config/hosts        /etc/
+COPY ./config/resolv.conf  /etc/
+COPY ./config/dnsmasq.conf /etc/
 
-# Start Container
-CMD systemctl restart dnsmasq
+## Reload Service
+RUN service dnsmasq restart

Config/DNS/docker-compose.yml

@@ -10,6 +10,14 @@ version: '3'
 services:
   app:
     build: ./
-    restart: always
+    restart: on-failure
     ports:
       - "53:53"
+    command: |
+      /bin/sh -c '
+        while :
+        do
+          sleep 8h & wait $${!}
+          service dnsmasq restart
+        done
+      '

Config/DNS/run.sh

@@ -0,0 +1,47 @@
+# 2023-07-29
+# Config/DNS/run.sh
+# Fix common issues when trying to run this container.
+
+function stop-service {
+	service=""
+	if [[ -n $1 ]]; then
+		service=$1
+	else
+		echo "ERROR: A parameter was not provided for stop-service, aborting."
+		exit 1
+	fi
+	if [[ -n $2 ]]; then
+		echo "ERROR: A second parameter to stop-service is not expected, aborting."
+		exit 1
+	fi
+	systemctl disable --now $service &&
+		echo "$service stopped successfully!" ||
+		echo "$service was not found, no problem."
+}
+
+echo -e "\n*** Turn off any local DNS programs ***"
+# These programs use port 53 but this container needs to be able to listen on it.
+stop-service systemd-resolved
+stop-service dnsmasq
+
+echo -e "\n*** Create a working DNS file ***"
+# Allows the domains needed during the docker pull/build to be accessed.
+if [[ ! -e /etc/resolv.conf.save ]]; then
+	# Save the existing file if a backup does not already exist.
+	mv /etc/resolv.conf /etc/resolv.conf.save
+fi
+echo "nameserver 1.1.1.1" > /etc/resolv.conf
+
+echo -e "\n*** Start the docker container ***"
+docker compose down
+docker compose build
+docker compose up -d
+
+echo -e "\n*** Now use the local process for DNS ***\n/etc/resolv.conf:"
+echo "nameserver 127.0.0.1" > /etc/resolv.conf
+echo "nameserver 127.0.1.1" >> /etc/resolv.conf
+cat /etc/resolv.conf
+
+# Finish
+echo " "
+exit 0

Config/DNS/undo.sh

@@ -0,0 +1,44 @@
+# 2023-07-29
+# Config/DNS/undo.sh
+# Easy way to stop using this container.
+
+function start-service {
+	service=""
+	if [[ -n $1 ]]; then
+		service=$1
+	else
+		echo "ERROR: A parameter was not provided for start-service, aborting."
+		exit 1
+	fi
+	if [[ -n $2 ]]; then
+		echo "ERROR: A second parameter to start-service is not expected, aborting."
+		exit 1
+	fi
+	systemctl enable --now $service &&
+		echo "$service started successfully!" ||
+		echo "$service was not found, no problem."
+}
+
+echo -e "\n*** Stop the docker container ***"
+docker compose down
+
+echo -en "\n*** Restore the DNS file "
+if [[ -e /etc/resolv.conf.save ]]; then
+	echo "from backup ***"
+	cp /etc/resolv.conf.save /etc/resolv.conf
+else
+	echo "with Cloudflare ***"
+	echo "nameserver 1.1.1.1" > /etc/resolv.conf
+	echo "nameserver 1.0.0.1" >> /etc/resolv.conf
+	echo "options rotate" >> /etc/resolv.conf
+fi
+echo "/etc/resolv.conf:"
+cat /etc/resolv.conf
+
+echo -e "\n*** Turn on any local DNS programs ***"
+start-service systemd-resolved
+start-service dnsmasq
+
+# Finish
+echo " "
+exit 0