The wiki here doesn't have entries for OK or TX. From moped2.org, it appears that you can ride every road except "high speed highways" or of course interstates, in OK. If the road has mailboxes, driveways, and side roads connecting right to the road it is probably not a high speed highway.
In TX you are equivalent to a bicycle so bike lanes should be allowed. In TX you must stay as far right as practical, but still in the travel lane. Other states give a moped the right to the whole lane, but it is probably still safest to stay as far right as practical. That does invite cars to pass you with (hopefully) inches to spare but also when someone isn't paying attention and comes flying up behind you, it will be easier for them to miss you.
Riding on the shoulder probably won't get you a ticket unless it's actually a sidewalk, not a shoulder, or you're doing it to get an advantage in passing traffic.