Some apparent misconceptions in the Q:
First, issuing money is complicated thing. Straight "printing" or increasing the monetary M1 mass (be it electronic) is not how central banks normally fight a recession or downturn. The normal mechanism is to lower the interest rate. It's only when you can't do that (the famous "zero lower bound"--ZLB) than more creative methods like quantitative easing have to be used.
The general danger with just printing money as economic stimulus is runaway inflation. Which Argentina is experiencing right now, and I think Ecuador was experiencing too when they decided to dolarize. Or when Bulgaria decided to peg their currency to the Deutschemark etc. (aka currency board, which is a half-way measure to just giving up your currency completely. You still retain the possibility to unpeg somewhat more easily, although there's a more complicated discussion relating to credibility of the peg.)
At ZLB, quantitative easing can be done is such a way as not to cause inflation. This is possible because inflation is not just more money being available or stored, but more money circulating (the famous velocity of money matters--if agents just "sit" on their money, it doesn't matter how much money they have or is given to them "out of nowhere"--prices [aka inflation] can't increase if nobody is buying anything extra. Which sometimes happens due to psychology/expectations.)
Fourth, most economic agents get money via M2 rather than M1, i.e. they get their money from commercial banks, which typically produce much more of it than M1, i.e. they're allowed to multiply the central bank money to some degree. (In fact, in the US, the Fed is strongly opposed to any bank that doesn't do that.) These banks are however sensitive to the central bank interest rate.
Fifth, most developing (or "2nd world") economies much more seldom reach the ZLB (in their own currency). This is because "1st world" currencies (and not just that of the US, but e.g. the Swiss franc too) act as an attractor when there is an economic crisis, but other currencies do not. So there's almost no reason for developing countries to "print money" (in their own currency) as a stimulus. It's only done there in an inflationary manner because they (often enough, alas) fail to otherwise have sound economic policies.
So if you give up your currency entirely, you give up a fairly useful lever, but not just/primarily because of lack of printing though. Lack of own printing is actually much more often the benefit.
Finally, currency substitution (dolarizaion included) is pretty close to a monetary union. There is an elaborate theory when the latter makes sense. One of the cases where it doesn't is if there's high risk of "asymmetric shocks", e.g. having an economic crisis/downturn in one part of your union, but not in another. So, if say the US economy experience a downturn at the same time as Ecuador's then that's that much less of a problem [for Ecuador] than if they happen at different times. And the level of synchronization between economic cycles is fairly dependent on the level of economic integration in broader terms.
There is a benefit [of sorts] for a state to print money though, even in inflationary manner: it devalues the state's own debt in their own currency, essentially giving itself a cut on their own debt without the (domestic) creditors having to explicitly approve of that, and even without passing legislation that could otherwise force them to agree to the cut. You'll note that Argentina has a high level of debt... However, in other corners inflation is also considered a bad way to approach debt, although historically speaking even the US has done some of that.
OTOH dolarization has also been desrcibed as a crutch for not being able to legislatively act against high deficit spending directly.
There is one more thing to keep in mind though: some countries, including Argentina experience effective dolarization [to some degree] even when they have their own official currency, if a lot of the population tries to avoid using it. That has nearly as many downsides as doing it officially. This includes the fact that people literally stashing dollars under their mattress means fewer reserves under the control of the authorities, so less knobs for them to tweak.
In a highly dollarized economy, the external credibility of the national currency is usually already largely compromised. And since the U.S. dollar is an international “reserves money,” economic agents who hold a large amount of dollars have the means to react to a temporary external shock. Thus, the main difference in the status of “reserves” seems to be that these “reserves” are in the hands of agents, rather than in those of the monetary authorities. [...]
In a highly dollarized economy, the foreign currency component of broad money cannot be directly influenced by the monetary authorities. Money supply in the economy is not determined by the monetary authorities but by the behavior of agents holding both foreign- and domestic-currency-denominated assets, including cash. As money supply in the economy becomes endogenous, the authorities may not be in a position to fight inflation by tightening domestic money supply in an appropriate manner. Based on empirical evidence, Hoffmaister and Végh (1995) assert that in Uruguay (a highly dollarized economy), dollarization may have severely hindered the effectiveness of monetary policy.
(Uruguay is/was not officially dollarized.)
Somewhat interesting, Uruguay has pursued a policy of de-dollarization since then, partially successful [in that regard]:
[D]eposit de-dollarization in Uruguay has had mixed results. The dollarization of fixed-term deposits fell from 91% to 51% between 1998 and early 2020. However, the dollarization of demand deposits has persisted at approximately 80% since 1999.
El Salvador is a paradigmatic case for official dollarization: OD was adopted in quiet times,
without inflationary or exchange rate pressures, with an eye set on the traditional theory of monetary unions
focused on the dilemma between trade gains (in a country with strong ties to the United States both in trade
and through labor exports via remittances accounting for more than 15% of GDP) and economic cycle synchronicity. The other official dollarization case in the region, Ecuador, is more problematic: it resulted from a
desperate attempt to put an end to a devastating currency crisis, in a country soon to become a commodity
exporter with limited access to international finance.
The analysis for El Salvador in that paper concludes that they haven't benefited much if at all from OD, although this conclusion is somewhat confounded by structural factors in El Salvador's economy.
OTOH, in crisis-hit Ecuador, it was apparently more beneficial, at least in the short run:
The first issue worth highlighting is that, like the Argentinean currency board (de la Torre et al., 2003) and as
argued by economic theory, dollarization helped to contain and reduce inflation, albeit at the expense of a
somewhat higher volatility of the economic cycle.
And Venezuela has experienced "spontaneous" dollarization this century, even somewhat applauded by its anti-US president, perhaps surprisingly.