Being an American ought not be a problem.
It's not like Iran, etc.
In my experience you are able to get visas with less difficulty than Brits.
The tricky bit is obtaining the visa without a CdH (proof of lodging or an escort itinerary).
See what the US consulate requires.

https://sahara-overland.com/algeria-3/

Technically, in the north (not Sahara) you don't need an escort but very few manage to get out of the border without one and it's exceedingly rare anyway.

Another thing to know is that northeast Algeria and parts of the Tunisian border are where jihadists hang out. Nothing like on the scale of the Sahel and roadblocks will stop you wandering into dodgy areas.
You could cross high - near the Med (never done it), or low at El Oued Nefra.
Avoid both sides of the border area in between.

As a seamless transit from Morocco is not possible, you have to wonder if the expense (2 x €500+ ferries + escort?) for a couple of days border-to-border is worth it.
Better to make the most of the visa and cough up for an escorted ride down to Djanet and the famous sign beyond (below, 2018). Allow at least 10 days and €2000 for the escort. Fuel is among the cheapest in the world. other costs minimal too.

Having travelled all over the south for nearly 40 years with motos, 4x4s and camels, I couldn't agree less with PanEuropean's negative impression of Algeria or comparisons with its neighbours which I also know well.
Most Sahara travellers would feel the same way.
It's just a huge shame travel down there is so restricted at the moment.

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Morocco Overland 4 (2025) • Trans Morocco Trail

Hello Chris:

Maybe Algeria has changed since I worked there in 2000 - 2001. I was based in numerous locations all over the country (Hassi Messaoud, Ouargla, In Salah, In Amenas, Ghardaia), and had the opportunity to see pretty much all the country in the course of my work as an aircraft pilot.

I found the government to be insufferable - in particular, customs & immigration - and I didn't see any geographic features in Algeria that can't be found in more hospitable countries such as Morocco & Tunisia.

At the time I was there, there was a lot of conflict. We used to remark that we didn't need to look for the (non-existent) windsock at the airports - we just had to look and see which way the heads were rolling across the runway.

Sorry to disagree, but in my opinion there are much more pleasant places to spend time in when in North Africa - Morocco, former Western Sahara, Tunisia, and Libya (if things ever settle down there). All of which are countries I have both lived in and worked in.

Michael