Fridays (especially in the evening) and Mondays (especially in the morning) are the hardest times to find cheap fares, as trains tend to get full of business passengers at those times (and lots of people want to get away for the weekend too)
Advance fares are usually made available 12 weeks in advance but, despite your dates only being 9 weeks from now, there aren't any Advance fares currently shown for your chosen dates. (There are sometimes periods, when the system is being restructured, where advance tickets only become available a month or so ahead of travel). So it might be best to wait awhile before looking for Advance fares for your dates.
Even so, travelling at either side of a weekend always tends to be pricey, even using Advance fares. Looking at an earlier weekend (Fri 4 Oct to Mon 7 Oct) to get an idea of what you're likely to find when Advance fares become available to you, suggests that the majority of services will probably be priced at £33.80 each way ( = £67.60 return) without a Senior Railcard. However if you can travel out in the late evening (2017 ex Newcastle, arriving in Lincoln at 2340 after an hour's wait for a connection at Newark North Gate) the fare will probably only be £17.00 single without a railcard.
Similarly, if you can wait until 1932 for a train back from Lincoln (arriving in Newcastle at 2239, with 49 minutes to connect at Doncaster) you'll probably only have to pay £17.00 single without a railcard too. (It's even cheaper, at just £12.40, at 2031 but that requires two changes and doesn't get you into Newcastle until 0043).
With a Senior Railcard you should be able to get fares of £22.30 each way (=£44.60 return) with a wide choice of services. The 2017 outbound service will probably be offered at £11.20 single. The 1932 back will also probably be available for £11.20 single, with the 2031 service being priced at £8.20.
So look again in a few weeks time, to see if Advance fares are being shown for your chosen dates. There's only one site you should ever look at:
https://www.nationalrail.co.uk
(Sites like The Trainline can't offer any cheaper fares, as they're accessing exactly the same database. They actually charge MORE - because of their booking fees - than going direct to National Rail).