Depends a lot on who your audience is, since they'll be singing along, but here are some songs I use for July 4th programs. Mine are mostly at retirement homes, but there are often several generations present, since it's a party setting.

1. Yankee Doodle - 3 regular verses (See DT) plus the "stuck a feather in his cap and called it macaroni" verse, since everyone knows it.
One verse I use, which I think is still relevant but not in the DT, is:
And there we saw a thousand men as rich as Squire David,
And all the money they did waste, I wish it could be sav-ed.

2. The House I Live In
Original lyrics
lyric portion in DT, from Earl Robinson recording for "Songs For Political Action"
Here these lyrics sung by Paul Robeson
The first few verses, as sung by Frank Sinatra:
Original Frank Sinatra recording
Frank Sinatra film short with The House I Live In (This song starts about 6:50 min into the 10 min film)
sung by Patti LaBelle

Another one I use is "A Hymn For All Nations" (music: Finlandia). The first 2 verses (the non-religious ones) are HERE, in the DT.

Then there's The Power And The Glory (Phil Ochs)

Pastures Of Plenty (Guthrie)

And, of course, one of the most popular ones is God Bless America.
Especially if you include the introductory verse, you can see that this song is a prayer of gratitude and for guidance "from above," not a chauvinistic song.   (Yes, it's disputable how "free" we have been at times and are now, but I still find this song one of the better "patriotic" American songs.)

Wasn't That A Time? (The Weavers) - probably one of the best. And there are other verses, plus it's easy to make up new ones.

THE LIBERTY TREE (DT) - Not an easy singalong, but interesting as a trad. song.

There's also Sousa's Stars & Stripes Forever - I sing the lyrics, such as they are, but mostly I do the whole march on the kazoo. I do it on kazoo first, then singing the verse:
"Hurrah for the flag of the free,
May she wave as our standard forever,
The gem of the land and the sea,
The banner of the right.
Let despots remember the day
When our fathers (forebears) with mighty endeavor
Proclaimed as they marched to the fray
That by her right and by her might she waves forever."

Then immediately back into the kazoo solo (including the piccolo part of the march) and crescendoing into the final verse (with everyone singing along):
"Be kind to your web-footed friends,
For a duck may be somebody's brother.
Be kind to your friends in the swamp,
Where the weather is cold and domp.
Well, you might think that this is the end -
Well, it is!"