Moving Nintendo eShop games from a 3DS to a New 3DS

The short answer is no, you can't transfer individual eShop games.

From Nintendo's 3DS General Information page:

Can the downloadable version of a Nintendo 3DS game be transferred to another 3DS console?

It is not possible to transfer a specific software title to another Nintendo 3DS console. However, it is possible to perform a System Transfer and transfer the entire content of your console to another. For more information please refer to the System Transfer section of this Website.

I had to dig a bit to find this information, it is under 'Nintendo eShop Information -> Downloadable Retail Software - Frequently Asked Questions'. Also keep in mind this tidbit regarding System Transfer, under 'Transferring Games':

How Many Times Can I Transfer Content from One System to Another?

Data can be transferred up to five times. A system cannot perform a system transfer multiple times within a set period.

Unfortunately, as you have noted, without the original system to transfer to, you are likely without hope. 3DS titles are tied to console, not to any ID (Nintendo Network ID or otherwise), so without the console, you can't do a whole bunch. If you had the console, you'd be able to do things differently, but as it stands, there isn't much you can do.

There are stories online where people say you can contact Nintendo to get your stuff back, but since Nintendo themselves seem pretty hush on the matter (their website states that you can only do things through transfer), I don't know how lucky you will be. Club Nintendo keeps a sort of record, in a way, so they might be able to help that way.

I would also keep in mind that stores usually wipe any used consoles they receive, so even if you can get to the store, it is not necessarily true that you can even get your stuff back, even if they have your particular console still, as doing a format wipe of the SD card and the system means that your information will also be gone. This prevents the next owner from being able to essentially get free games by downloading what you had already purchased.