I started typing a reply about checking for USART/USI/USCI differences and suggesting looking at the MSP430Ware examples. (Nowhere near as detailed as Katie's, of course.)
However, then I checked the spec pages and saw they were both USCI. I didn't even notice the subtle eUSCI change! I guess that reinforces the point. It's usually the first thing I struggle with and starting from a working example usually speeds things up for me. Going back to the USART on an AFE253 was particularly painful recently.
1. Your GPIO setting looks incorrect to me. You are using P2.5 + P2.6 for UCA1TXD/RXD. However, looking at the FR5969 datasheet http://www.ti.com/lit/gpn/msp430fr5969 on p. 88 table 6-52 I see this for the pin setup:
You need to have P2SEL1 bit 5 and 6 set to 1, and P2SEL0 bit 5 and 6 set to 0. Right now, it looks like your code does the opposite:
P2SEL0 |= RXD + TXD; //Setup the I/O //P2SEL1 |= RXD + TXD;
So switch which line is commented out.
2. You are now using FR5969. This device has an eUSCI module (enhanced USCI) rather than USCI that was present on G2553. There are a few differences (for more detail please see the USCI to eUSCI migration guide http://www.ti.com/lit/pdf/slaa522).
One thing is the interrupt vector is different on eUSCI so this needs to be changed. It should be the USCI_A1_VECTOR, and you should also modify the ISR to have a switch statement to handle and respond to the correct values for UCA1IV because this is different. You may want to look at some FR5969 code examples for a demonstration of this.
Another thing is the baud rate generation. The eUSCI module provides a few more bits to get you better granularity and accuracy for generating baud rates - please check the FR5xx/6xx user's guide http://www.ti.com/lit/pdf/slau367 Table 21-5 for a list of recommended settings to generate different baud rates, and note the slight difference - you should change your code to match this (notably UCBSx = 0x49).
There may be other things to change, but hopefully these tips will help!