It sounds like fit and fitness issues to me. The drift from side to side has to do with core muscle strength that balances your body mass over the three contact points (saddle, pedals, handlebars). Are you shifting your body weight toward the downstroke? You'll need to maintain balance, keeping everything from the hips upward stationary while pedaling.

Also, visit a shop and ask them to check whether your seat height is a little high. This may cause your pelvis to shift or wobble back and forth. This will cause an imbalance.

Since your Nashbar hounder is single speed, you also have the issue of cadence and strength. You will be outside of the optimum power curve at times, so it'll feel like you're pushing hard sometimes, and that you can't keep up at others. For the times when you're straining, make every effort to keep your upper body quiet.

Learning to come out of the saddle briefly to add torque will help you. This isn't easy for me to explain, but you'll raise your bottom off the saddle and allow the bike to lean away from the downstroke while you pull with your arms. This motion will alternate side-to-side, and can give you brief bursts of additional power - at an energy cost to your body, of course.

The best advice is to check your fit and keep at it.

PG