Sounds like you have the same problem I've been dealing with this whole summer.
$2K later and it's fixed by replacing the trans.
In my trans I had was a bolt on the shift selector that broke. The broken bolt is what gave me problems pushing the shifter side to side. Note, when you push your shift lever side to side you are lifting (push shifter right for 3, 4, 5 & R) and lowering (rest position for 1 & 2) the shift selector in the trans. The broken bolt some how hung up left right movement but after a while shook loose but I was stuck in 4th.
At first I retap the hole and put in a larger hardened bolt;
nevertheless, I had the same failure once again in about 3 months.
From Dealers and Terry Haines I've learned there is a 2001 Shift Selector fix for this issue.
Early next year I plan to have Terry do the 2001 shift selector up date and Quaife.
Anthony B. Davis
2Deep2 Silver SVT
2002 IASCA SQ Ult Ama 601+ World Champion runner up
2002 SLAP dBQ Champion

2000 USAC SQ World Champion
2002 USACi SQ World Champion