You mentioned "a fee" you could pay to re-new your visa? ... I think it was mentioned in first post? Something like $290? (or so?)

That might end up being best/cheapest option for you. Dunno?
Staying in Mexico (or anywhere) over 31 days will surely be more than
$290 fee, yes? ... or if you fly somewhere, even more spent.

Or ... as you say, you could just leave and spend your money elsewhere. Surely would not blame you ... and Don't worry, no one will miss you.

Euro trash tourists were tripping over themselves last week in Yosemite, Sequoia and Death Valley parks when I passed through. I heard every language but English! Swedes, Danes, Brits, French (LOTS) ... and of course many Swiss and Germans. Dozens on rented Harley's. "It's SO CHEAP", they all told us every time we asked. Strong Euro, weak dollar means the High Life for Euro tourists in USA, no?

Sorry for the F'd up Migra rules here, our govt. is in Full Flight Panic and Fear mode now since our "new" war. Two steps away to going full Fascist.
Never ending war brings on such policies. The border area is now a War Zone. Use caution. Trigger happy Nut Jobs. (damaged robots from Iraq and Afghanistan)

But also consider that tens of thousands of "tourists" who come to the USA and WORK. They often pay no taxes (Cash jobs). I don't mean our 20 million (or so) hard working undocumented folks from all over Latin America, but rather Euro White folk with education and skills taking HIGH PAYING jobs that Americans could do. Mostly in IT and Tech areas.
Thousands come in on Tourist visas ... but work "side jobs".

Can I do that in Germany?

If an American wanted to work anywhere in the EU ... well, it's nearly impossible. (I know, I tried it.) But I was able to get LEGAL work permits in Mexico, El Salvador and Argentina!!

What ever you do ... safe travels, good luck!