Took a look at your save and have a few tips for you...hope they help.
Firstly, yes, your situation is not great, but hey, it’s your first attempt at prince and all part of the learning curve.

You're at that classic stage where you've grown through war and your economy is now feeling the strain. This is actually quite normal with 7 cities as you move up the levels and is partly due to the higher maintenance costs you're now paying. In your case, the problem is exacerbated by your lack of happy resources, which is limiting the number of tiles you can work in each city. Taking Ragnar's capital would've been ideal (as would’ve TGL) but it wasn't to be. Now you've to re-group and ready yourself for the next war.
As it stands, there are a couple of reasons why you're labouring. The first, as you've noted, is the lack of currency. No, your move to tech alphabet was not idiotic given how little commerce some of your cities are generating. However, this point does highlight a problem: some of your cities are barely generating enough commerce to cover their maintenance costs, so you need to favour working commerce heavy tiles. Perhaps it was also worth razing Haithabu given that it doesn't grant you a wonder, key resource, or provide much commerce? In addition, I’d be inclined to ask whether it was worthwhile teching monarchy before alphabet. Admittedly, HR raises the happy cap but, as maintenance costs rise with the playing levels, the ability of alphabet or currency to help you recover from expansion becomes ever more important. With either researched, you’d have been able to subsequently tech to monarchy (or anything else such as CoL) by building wealth or research as necessary. In the meantime, IMHO, the happy cap would still have allowed you to take on Ragnar by (i) whipping troops (ii) controlling regrowth and whip anger by running science specialists as needed and (iii) chopping every tree in sight. The last point is important since, if I recall, there are currently 7 forests still standing in your borders and you can chop them all without impacting health since they’re outside all BFC’s. That’s quite a few soldiers which would really have helped in the war.
Now, since we can only change the future (unless you replay an earlier save of course

), let’s look forward. With that in mind, my suggestions are as follows:
Tech path: currency, CoL then depends on how things are with Rags.
Talking of Rags, my first priority at this point would actually be to whip some spears pronto in Heliopolis, Elephantine, Haithabu and Uppsala to protect your border cities. (NB: walls would’ve been ideal but I note you don’t have masonry or stone.) Whilst it looks like you should be able to out-produce him having acquired a few of his cities, your problem at the mo’ IMHO is that you’re defending two borders against chariots and HAs – both very fast units that can attack your peripheral cities within a turn or two. Spears are the order of the day IMHO. I’d also move Memphis’ axe to the front line ASAP. Once done, its libraries as needed and a WB.
To speed research, I’d also suggest using a few more science specialists where possible. Assuming you build / whip spears as mentioned, you can work the pig, 2 cottages and 2 scientists in Heliopolis, and two scientists in Elephantine and Thebes. Having built the Moai in Memphis, work coastal tiles here to get commerce. For future reference, it’s worthwhile whipping a granary (eg. in Uppsala) when the food bin is at most half full to speed regrowth. Whip your two settlers in production next turn for 2 pop and build military as you regrow, trying to work cottages as you do. After finishing the lighthouse, Alexandria should invest one turn in a galley, whip it for two pop and then build / whip an archer or two for your settlers. I’d split your workers between (a) chopping remaining trees for military and (b) clearing the jungle on the island to the W and mining the gems for your new city...time for automation to cease IMHO.
Hope this helps.
BTW, in light of your following comment,
Sword rush seemed more viable option, but given that I was in an uncomfortable situation I probably made a lot of stupid mistakes, because I just didn't know what to do.
don’t worry, that’s precisely when these forums can help.

With that in mind, if you’re looking for further tips, look in on Shafi’s current and / or previous HUI games played on monarch (or even Sis' ALCs) – hopefully, they'll help you climb even further than prince.
