You can use shortcodes of other plugins in Ad Inserter.
However, in your case you would probably like to replace old shortcodes with Ad Inserter ones.
There are at least two possible approaches.
1)
The easiest way to insert ads on posts is to use automatic insertion. This is also the best approach as you don’t have to modify posts or pages to insert or change ads.
https://adinserter.pro/documentation#settings
For example, you configure Ad Inserter to automatically insert ads before the second paragraph and after post content.
Some common settings
In this case you would need to remove all old shortcodes.
2)
The second approach is to replace old shortcodes with Ad Inserter ones.
This is not a good approach as you’ll have to manually insert shortcodes into every new post.
3)
Of course, combination of both approaches described above is also possible.
To replace old shortcodes I recommend Search Regex plugin.
For example, to replace [ad name="HTML-1"]
with [adinserter block="1"]
you should use

View post on imgur.com
First click on Search to see all occurences of the old shortcode. Once you are happy click on Replace to replace them.
PLEASE NOTE: after replacement there is no way back! Make sure you have a backup of the website (or database) in case something goes wrong.
To delete old shortcodes simply replace them with empty string.
When using Ad Inserter shortcodes don’t forget to enable them (button Manual).