I starch my fabrics. If I'm working on a quilt that has tiny pieces I heavily starch the fabrics. I don't use BEST PRESS, canned starches, or sizing. I purchase STA-FLO in a large bottle and mix it myself and put it into a spray bottle. I use a 50/50 mixture. Yes, I do get "flakes" but those brush off easily. You must iron the fabric dry. I do not use steam in my iron when I use starch. I get a much better starch set this way. Here is what I would do in your situation:
Starch fabric before cutting
Make strips, starch, iron in the tube
CAREFULLY remove stitches between selected pieces to prevent stretching
If the strip is narrow I would starch again
When working with small pieces, instead of ironing the length of strip I would iron side to side and move strip
I find that by starching my fabrics and piecing as I go along I get more accurate seams and the fabric is easier to work.