You had me largely convinced on Mrs Dashwood writing only one letter here.
The only reason I remain abit unsure is it may have been good manners for Mrs Dashwood to write a seperate letter to thank Sir John for his generosity... But I have been unable to find anything on letter etiquette in JA's time on this point.

I doubt Mrs Dashwood's intention was to give Sir John extra expense, but she has a warm and impulsive temperment.
'She instantly wrote Sir John Middleton her acknowledgement of his kindness, and her acceptance of his proposal;' (ch.4)

Also, even if Mrs D. did not intend to take the offer, she may've felt obliged to write to thank Sir John instead of remaining silent- and Sir John would pay this postage cost.