Hi Allison,
Here is my answers on top of your question. Hope this helps.
How many servers do you have other than production?
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For my experience you should have at least 2 environment DEV, PROD(Always good if you have another instance UAT) for on premise.
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How do you organize your solutions to keep everything organized and nothing getting into production before it's supposed to?
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I believe you are aware about solution patching I would always recommend OOB feature instead of manually separate the solution component in terms of bug fixes or new features.
[View:https://www.inogic.com/blog/2016/01/solution-patching-in-microsoft-dynamics-crm-2016/]
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What are people's recommended practices around their own internal customized solution when rolling out bug fixes and new features?
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Depends on your customization ,If project size is small then I would recommend to maintain single unmanaged solution in DEV environment where you should keep your all customization. If your customer wants to change anything on top of your changes you can import unmanaged solution in PROD environment.
For bug fixes I would suggest to use out of the box solution patching so that you can easily uninstall the patch for any kind of issues without impacting the base solution.
And for new features or module if the entities and functionality are fully different then create new unmanaged solution and do your changes once done you can separately import into prod. Now if the entities are related to the base solution which you delivered in PROD then you can go for solution patching again.
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Do you use totally new servers, or just different organizations?
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I would always prefer having new servers to avoid down time.
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Do you do anything different step-wise for bug fixes vs features?
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I am not very much clear your question . It's depends on the fixes and features.
