Opposite side turn signal blinks on 2001 town & country lxi

Likely a bad ground, but could be a fuse(s), or a failed multifunction switch, or BCM. It requires circuit testing to identify the cause.

It could be the lamp control module !
Headlamps
Battery positive voltage (B+) is applied at all times to the lamp control module (LCM) through the underhood fuse block HDLP fuse.
When you place the headlamp switch in the HEAD position, the headlamps input to the LCM is grounded. The LCM then applies battery positive voltage (B+) through the underhood fuse block HDLOBML fuse and HDLOBMR fuse, which feed the left low beam headlamp and the right low beam headlamp respectively. When you activate the headlamp dimmer switch by pulling on the multifunction lever until the switch clicks, the high beam input circuit to the LCM in grounded . The LCM then removes battery positive voltage (B+) from the low beam headlamps and supplies battery positive voltage (B+) through the underhood fuse block HDHIBML fuse and HDHIBMR fuse, which feed the left high beam headlamp and the right high beam headlamp respectively.
Check this fuse in the left under hood fuse box first. HDLOBMR Fuse 18 10 Amp. RH Low Beam Headlamp . If the fuse is good check for B+ voltage with the low beams on . Use a test light or volt meter . If you look on top of fuse there are two metal tabs ,check there for B+ voltage , If no B+ voltage the LCM - lamp control module isn't supplying any .
Lamp Control Module (LCM)
Under the left side of the IP, to the right of the steering column
Turn Signal Flasher
Under the left side of the IP, to the right of the steering column
Lighting Systems Component Views in Lighting Systems
Lighting Systems Connector End Views in Lighting Systems
Turn Signal/Hazard Switch
In the left front of the passenger compartment, in the steering column

Headlamps both may be burned out

replaced the dimmer switch for it is shorting out for the low beam circuit

If you have separate bulbs for the low and high beams(4 light system), if you pull the turn signal and hold it, all 4 lights will stay on until you release the turn signal. You need to replace the bright bulb on the right side and the low bulb on the left side. If your headlights only have one bulb on each side, then you will have to replace both of them to get everything to work.

multi-fuction switch (headlights)

Right headlight sounds blown to me. Other problem needs to be looked at.

most of those issues i have dealt with are the connector on the turn signal switch or the switch itself

Ill guarantee you have an electrical short on the left side of the headlight switch,circuit.From here its immpossible for me to go any further.Good Luck.,Get A Wiring diagram.