Mass Transit - only single Consumer "handler" is processing messages

will help you Cleanly shutting down a MT service will wait for any messages in process to completely finished. Why sometimes it takes a little while for a service to shutdown. Shutting down the service is the best way to handle this, you are sure MT is not pulling any new messages.
If your sagas are serialized to a backing store, e.g. NHibernate, then the state will be saved until the service is restarted and the sagas will pick up in the state they were left after the last message was processed. You should be in good shape. We do this all the time for any maintenance periods.

This might help you RabbitMQ consists of exchanges and queues.
Exchanges are created by MassTransit when publishing messages, based on the message types. Publishing does not create any queues. Queues are where messages are stored for delivery to consumers.

will be helpful for those in need Add a reference to the MassTransit.SimpleInjector package, and then in your receive endpoint, configure the consumer using:

cfg.ReceiveEndpoint("queue_name", e =>
{
    e.Consumer<AddMyCommandConsumer>(container);
});

container.Register<AddMyCommandConsumer>(Lifestyle.Scoped);

container.Options.DefaultScopedLifestyle = new AsyncScopedLifestyle();

hop of those help? You shouldn't need to have the consumer on your client-side API. The only thing I can think is that you didn't have the topology setup right.
Make sure the queue exists, in advance, since functions don't create the queues. Configure the URI address for the queue, so that your web API can specify it on the AddRequestClient call as an argument (_sb://host..../input-queue). Profit!