VHS should be captured with a QUALITY machine using either a VC-500 or Hauppage device, both USB. Other devices are mostly junk.
VHS is interlaced and should be captured thusly.
VHS should also be captured as uncompressed AVI to greatly reduce noise, artifacts from lousy MPEG compression.
Each time you convert you lose color fidelity. Other compressed formats like MPEG or MP4 are the final delivery codec and should only be used after all corrections have been made, color, stabilizer, etc.

If you want to really go full, you install PD as a 32-bit app, this allows for uncompressed AVI interface to appear and a whole new world openes up! Do you have an older version of PD? I use V12 for my uncompressed lossless capture. Has to be 32 bit.

You can also download STOIK capture utility, free, and get the benefit of the AVI codecs without resorting to a 32 bit PD.

Lossless codecs and the VFW interface are all over the internet. I use UTvideo codec because it works with no issues in PD and I can fix colors, upsample, all the good stuff.

A Time Base Corrector is also very important, but expensive. Mine is $400 used.
An alternative to a TBC is a Panasonic ES10 or ES-15, they have an uncanny ability, used as a pass-through, to eliminate flagging and tearing, very much like a TBC, but can be bought used for maybe $150 or less.

It all starts with a quality VCR, what are you using, exactly? HP Envy Phoenix/4thGen i7-4770(4@3.4GHz~turbo>3.9)
Nvidia GTX 960(4GB)/16GB DDR3/
Canon Vixia HV30/HF-M40/HF-M41/HF-G20/Olympus E-PL5.
Tape capture using 6 VCR, TBC-1000, Elite BVP4+, Sony D8 camcorder with TBC.
https://www.facebook.com/BarryAFTT

I don't recall the VCR I am using. It is a SONY dvd-r/VCR combo. I am using a VC-500 for capturing. As I said previusly, when I watch the MPEG2 file created by PD14 from the capture, it looks fine and I don't have those interlacing problems. However, when I edit and convert to MP4 I get interlacing problems.

So you are saying PD14 doesn't allow you to capture to AVI unless you install as 32-bit and I should consider STOIK instead? If I ended up switching to 32-bit PD14, what would I be giving up? I have an old version of PD, but not sure if I still have the install files for it.

So, when I capture to AVI, can I then edit in PD14 and save to that H.265 HEVC MP4 format and have my file compressed without the interlacing issues? Basically, do I just need to keep the file format to something like AVI? Is there anything special I need to do when compressing the AVI file to MP4 to avoid interlacing issues?

Thank you for helping me on this!!

VC-500, excellent choice.
Sony player, better than many.
For lossless AVI, not many 64-bit consumer NLE programs, if any, can utilize lossless avi codecs, therefore I use 32bit anything, my choices are 32-bit PD, any version, or Stoik, a freeware.
Of course you must seek out and download lossless codecs such as Huffyuv, MagicYUV, Lagareth, UTvideo, and many others. I only use UTvideo as it has no issues within PD interface, I can utilize all the improvements and FX filters to repair the video, however, that may be specific to my setup, and you may be able to use other codecs.
Giving up 64 for 32 bit reduces the abilities of PD, that is why I capture with an old version installed as 32 bit, but edit in a newer version with full 64 bit.
Did your dongle come with PD? It might be a cut version, but still, if you click on the capture tab, choose AVI, and then click settings, you might find you already have the ability. Guessing.
This also allows for Divx capture, and actually, you can produce to Divx and lossless also.
Now, all that being said, I am not really an expert. I do very well when things go right, but real experts know how to fix when things go wrong! I am a little perplexed at your interlacing issue.
I suspect you are capturing MPEG2, from a ready-made PD profile, if staying with MPEG2, you should customize the profile to quality=7, check denoise, and bump up the capture to 12k. It defaults to interlaced, upper field first.
In my experience, with either MPEG2 or lossless AVI, I can tweak and produce a progressive file that plays fine. What software are you playing your files in?

Please note: The true experts are at DigitalFAQ, you can browse and search their forum, but they are Puritanical in their zeal to capture every last pixel, and mostly use scripting in VirtualDub to repair videos, I am not so pure. HP Envy Phoenix/4thGen i7-4770(4@3.4GHz~turbo>3.9)
Nvidia GTX 960(4GB)/16GB DDR3/
Canon Vixia HV30/HF-M40/HF-M41/HF-G20/Olympus E-PL5.
Tape capture using 6 VCR, TBC-1000, Elite BVP4+, Sony D8 camcorder with TBC.
https://www.facebook.com/BarryAFTT

Are you still having issues? Could you capture 10 (+ seconds of video and attach it here? HP Envy Phoenix/4thGen i7-4770(4@3.4GHz~turbo>3.9)
Nvidia GTX 960(4GB)/16GB DDR3/
Canon Vixia HV30/HF-M40/HF-M41/HF-G20/Olympus E-PL5.
Tape capture using 6 VCR, TBC-1000, Elite BVP4+, Sony D8 camcorder with TBC.
https://www.facebook.com/BarryAFTT

Is your visible interlacing only in Preview, or in the Produced new file? HP Envy Phoenix/4thGen i7-4770(4@3.4GHz~turbo>3.9)
Nvidia GTX 960(4GB)/16GB DDR3/
Canon Vixia HV30/HF-M40/HF-M41/HF-G20/Olympus E-PL5.
Tape capture using 6 VCR, TBC-1000, Elite BVP4+, Sony D8 camcorder with TBC.
https://www.facebook.com/BarryAFTT