No. After Scotland regains independence you’d be able to travel freely between Scotland and the continuing UK. What’s more goods would be able to travel freely too. But even more than this, with your EU Citizenship returned you’d be able to travel freely (and live and work) in any other EU country.

Firstly, as the Westminster Government admits itself [1], you’d still have British citizenship after independence. And more than that, your children would also be entitled to this (but not grandchildren living in Scotland – that’s a limitation applied by Westminster to anyone moving to another country) – see above.

Secondly, Scotland and the continuing UK are part of the Common Travel Area (CTA) alongside the Republic of Ireland. Again, most experts believe that Scotland would remian within the CTA after independence [2]. This allows freedom of movement and trade between countries across the Anglo-Celtic Isles.

Because this would open an EU border (between Scotland the continuing UK) it is likely that Scotland would negotiate an opt out of the Schengan agreement, as Ireland has done (for the same reason). This means you’d still need to show your Scottish/EU passport at the border when travelling to other EU countries but no visa checks are required – and at least you’d be able to use the “EU Citizens Only” lanes instead of queuing for hours as “Non-EU Citizens”.

Sources

[1] https://assets.publishing.service.gov.uk/media/5a7c2b08e5274a1f5cc763ac/scotland_analysis_borders_citizenship.pdf

[2] https://www.centreonconstitutionalchange.ac.uk/sites/default/files/2020-03/An%20Independent%20Scotland%20in%20the%20EU.pdf