Whenever you need to ask where to ask a question, the best thing to do is read the help center:
First, from "What topics can I ask about here?" in Stack Overflow's Help Center:
- a specific programming problem
- a software algorithm
- software tools commonly used by programmers
- practical, answerable problems that are unique to the programming profession
Next, from "What topics can I ask about here?" in GameDev's Help Center:
- game design (level design, gameplay, mechanics, etc)
- asset pipelines (creation, storage, editing, etc)
- game-specific programming issues (engine architecture, game-related APIs, networking, tools, etc)
- project management (testing, team management, scheduling, publishing, etc)
- game industry (careers, trends, technology, etc)
So, it would seem that programming games with monogame would be on-topic on both sites, but other development related questions would be more on topic for GameDev.
So how to decide where to ask programming questions? Well, that is entirely up to you. You are going to get a wider audience on Stack Overflow, but you are likely going to get people with a breadth of knowledge. You are probably going to find an audience that has more in-depth knowledge in game development on GameDev - so it is up to you to decide which audience you would prefer.
And there is nothing stopping you from asking the question in one location, evaluating the answers for a few days, then flagging it for migration if you don't think your answers are good enough.1
As Bart mentions in the comments, supporting the small site is an excellent idea. And if you don't get a response, you can go back and flag the question for migration to SO later.
1 - There is no assurances that just because you flag the post for migration, it will get migrated. A lot depends on the specific situation, the answers, how much overlap there is between the sites and how on-topic it is in both locations. You should ask the post in the location you think fits best and if the answers aren't what you were expecting, in terms of scope, then explain that when you flag for migration.
The other option would be to change the question to fit the scope of the site. The community frowns on cross posting on multiple sites, so don't copy and paste the exact question and do not do it at the same time. But if your post has not fared well enough and it has had sufficient time, and if you tailor the question to fit the scope of the other site and you should be able to post it on the 2nd site without too many objections.