Auto Increment Field in Django/Python
I came across this looking for something else and wanted to point something out:
By storing the order in a field in the same table as your data, you lose data integrity, or if you index it things will get very complicated if you hit a conflict.  In other words, it's very easy to have a bug (or something else) give you two 3's, a missing 4, and other weird things can happen.  I inherited a project with a manual sort order that was critical to the application (there were other issues as well) and this was constantly an issue, with just 200-300 items.
The right way to handle a manual sort order is to have a separate table to manage it and sort with a join.  This way your Order table will have exactly 10 entries with just it's PK (the order number) and a foreign key relationship to the ID of the items you want to sort.  Deleted items just won't have a reference anymore.
You can continue to sort on delete similar to how you're doing it now, you'll just be updating the Order model's FK to list instead of iterating through and re-writing all your items.  Much more efficient.
This will scale up to millions of manually sorted items easily.  But rather than using auto-incremented ints, you would want to give each item a random order id in between the two items you want to place it between and keep plenty of space (few hundred thousand should do it) so you can arbitrarily re-sort them.
I see you mentioned that you've only got 10 rows here, but designing your architecture to scale well the first time, as a practice, will save you headaches down the road, and once you're in the habit of it, it won't really take you any more time.
I have a table in a django app where one of the fields is called Order (as in sort order) and is an integer. Every time a new record is entered the field auto increments itself to the next number. My issue is when a record is deleted I would like the other records to shift a number up and cant find anything that would recalculate all the records in the table and shift them a number up if a record is deleted. 
For instance there are 5 records in the table where order numbers are 1, 2, 3, 4, and 5. Someone deleted record number 2 and now I would like numbers 3, 4, and 5 to move up to take the deleted number 2's place so the order numbers would now be 1, 2, 3, and 4. Is it possible with python, postgres and django?
Thanks in Advance!