To answer the question, there are two things going on.
One is the interference from the transmissions, which lowers the signal-to-noise ratio if you are on the same (or an overlapping) channel. The lower that gets, the more errors and the worse your performance. So to combat that, you can either use a different channel (1, 6, and 11 are probably the best to try, but you could just measure your performance on all of them and use whatever is best), or as you said force the use of a lower data rate (a lower data rate is more tolerant of low signal-to-noise ratios). Obviously there's a trade-off between having a more reliable connection with a lower data rate and having an unreliable connection with a faster data rate.
The second issue is channel reservation. Because of how 802.11 works, you can end up with a situation where one user is getting an unfair amount of bandwidth by basically telling the other users to be quiet. There are various settings which you may have access to depending on your router, such as the RTS/CTS threshold (try setting it to a very small value, may be better or worse depending on the situation), the backoff window size (which tunes aggressiveness), preamble (a long preamble helps when there is interference), etc.
So yes, just try different settings and you should be able to get an improvement.