Changing core systems is a project, and it should be planned like one. The biggest decision is timing: cut over away from fiscal year-end and away from your busiest season, so the people who know the numbers have time to check them.
Decide what moves. At minimum: open jobs with their budgets, cost to date, open commitments, billed to date, retainage receivable and payable, and approved and pending change orders. History can often stay in the old system as a read-only archive, with summaries brought over.
Plan a short parallel period where the WIP and pay apps come out of both systems and must tie. Differences found in parallel are cheap; differences found by an owner or a surety are not. Keep exports from the old system, and make sure your data stays yours: you should be able to export it from the new one too.
Phase it. Many contractors start with one area (for example, job cost and commitments on new jobs) and expand once it is trusted.