I initially voted to close this question because I think asking about risk free rates is off topic on quant se (and answered several times). However, I do not think there is a good alternative where one can ask questions that are deemed to be too basic here. Therefore, I will try to answer and go beyond just interest rates inputs.
Based on the question and your comments, I suspect you ask about listed equity option?
As mentioned in a comment, Treasury bonds aren't good proxy for a risk-free rate and usually not used. E.g. Bloomberg wouldn't even allow you to select them on all their derivatives pricers. See here for some details. You can find an example of a SOFR swap in this quant se answer. Vendors like Bloomberg will have market quoted swaps for all sorts of tenors and stripped curves to allow you to get the exact quote needed for any option tenor. If you do not have BBG (or anything similar) at work, you can try a public library or university library. Chathamfinancial has some basic tenors which should be sufficient to get reasonable values if you don't have access to a more complete data source.
There is no need to adjust for inflation. Finance is about nominal values and option pricing is all about hedging and replication, leaving only the risk-free return.
However, the risk-free rate is only a minor input detail when pricing American options. Generally, these options are price quoted anyways. You cannot simply input values in the Black Scholes equation because there exists no closed form solution for American options. Frequently, the Cox-Ross-Rubenstein Model - CRR is used, which is a recombining binomial tree. It is a particular case of an explicit FDM scheme.
There are many papers discussing alternatives to CRR - see for example Andreasen, American Option Pricing in a Tick from Saxo Bank or Bank of America Merrill Lynch where the author claims to be able to improve the efficiency of American option pricing algorithms by at least 4 orders of magnitude. This Risk.net article also discusses fast and stable American option pricing with FDM.
CRR itself has also been modified because the convergence of the binomial tree-based value to the limit is not monotone but rather oscillatory. This observation was the basis for a method developed by Leisen and Reimer (LR) (1996) to compute accurate results with a "minimum" number of time steps.
Voladynamics also uses LR. The link has a presentation that provides an excellent summary of the complexities of dividend modelling that arise when pricing equity options. Voladynamics comes in places where things get really difficult, namely trying to fit vol surfaces. You can find a description of the commonly used steps in this quant se answer.
My point was really that I would not spend too much time worrying about an appropriate risk free rate if you are just trying to get something working on your own for educational purposes. There are many more challenging and complex issues that need to be solved before you need to worry about the exact risk free rate to be used in the model.
