Is there a way to create a new budget with adjusted account carryover balances that are reconciled?

First off let's get some terminology correct. Even though you can filter a budget by accounts, the budget is category based. You adjust category amounts.

The second part if what is the "rollover". They aren't "transactions amounts" to be reconciled. They can add up to what you reconciled in the past, but they really have no direct connection to past transactions. For instance, one might decide to rollover all the gas prices for the year, because how much they spend on a given month varies, but on average they spending/budget a given amount. The shortfall in one month gets made up in the next month. But they might want to start the next year without any rollover and as such zero out the rollover in January.

Next on the rollover amounts. Quicken definitely has some quirks on the automatic rollover numbers it picks up, but that isn't to say that one can't override them with whatever they like.

The first thing to understand is that for any category you want to change the rollover balance on you need to turn on the rollover option. Click on this icon to turn it on:

Once you have done that make sure you have the Balance column being displayed, and then select the category balance amount for the month you want to change and you will get this popup.

You can select Edit to change the rollover amount.

And on the Actual column, it shows actual transactions or reminders (in the case of reminders they might be prorated depending on when they fall in the month and what day of the month it is for the current month). You can decide to show reminders in your budget or not.